Calculus Examples

$f (x) = x^{2} {(2 - 5 x)}^{3}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = {(2 - 5 x)}^{3}$ .

$x^{2} \frac{d}{d x} [{(2 - 5 x)}^{3}] + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 2 - 5 x$ .

Tap for more steps...

Step 1.2.1

To apply the Chain Rule, set $u$ as $2 - 5 x$ .

$x^{2} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [2 - 5 x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$x^{2} (3 u^{2} \frac{d}{d x} [2 - 5 x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.2.3

Replace all occurrences of $u$ with $2 - 5 x$ .

$x^{2} (3 {(2 - 5 x)}^{2} \frac{d}{d x} [2 - 5 x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.3

Differentiate.

Tap for more steps...

Step 1.3.1

By the Sum Rule, the derivative of $2 - 5 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x]$ .

$x^{2} (3 {(2 - 5 x)}^{2} (\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x])) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$x^{2} (3 {(2 - 5 x)}^{2} (0 + \frac{d}{d x} [- 5 x])) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.3.3

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

$x^{2} (3 {(2 - 5 x)}^{2} \frac{d}{d x} [- 5 x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$x^{2} (3 {(2 - 5 x)}^{2} (- 5 \frac{d}{d x} [x])) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.3.5

Multiply $- 5$ by $3$ .

$x^{2} (- 15 {(2 - 5 x)}^{2} \frac{d}{d x} [x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{2} (- 15 {(2 - 5 x)}^{2} \cdot 1) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.3.7

Multiply $- 15$ by $1$ .

$x^{2} (- 15 {(2 - 5 x)}^{2}) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 1.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (- 15 {(2 - 5 x)}^{2}) + {(2 - 5 x)}^{3} (2 x)$

Step 1.3.9

Move $2$ to the left of ${(2 - 5 x)}^{3}$ .

$x^{2} (- 15 {(2 - 5 x)}^{2}) + 2 \cdot {(2 - 5 x)}^{3} x$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Factor $x {(2 - 5 x)}^{2}$ out of $x^{2} (- 15 {(2 - 5 x)}^{2}) + 2 {(2 - 5 x)}^{3} x$ .

Tap for more steps...

Step 1.4.1.1

Factor $x {(2 - 5 x)}^{2}$ out of $x^{2} (- 15 {(2 - 5 x)}^{2})$ .

$x {(2 - 5 x)}^{2} (x \cdot (- 15)) + 2 {(2 - 5 x)}^{3} x$

Step 1.4.1.2

Factor $x {(2 - 5 x)}^{2}$ out of $2 {(2 - 5 x)}^{3} x$ .

$x {(2 - 5 x)}^{2} (x \cdot (- 15)) + x {(2 - 5 x)}^{2} (2 (2 - 5 x))$

Step 1.4.1.3

Factor $x {(2 - 5 x)}^{2}$ out of $x {(2 - 5 x)}^{2} (x \cdot (- 15)) + x {(2 - 5 x)}^{2} (2 (2 - 5 x))$ .

$x {(2 - 5 x)}^{2} (x \cdot (- 15) + 2 (2 - 5 x))$

Step 1.4.2

Move $- 15$ to the left of $x$ .

$x {(2 - 5 x)}^{2} (- 15 x + 2 (2 - 5 x))$

Step 1.4.3

Rewrite ${(2 - 5 x)}^{2}$ as $(2 - 5 x) (2 - 5 x)$ .

$x ((2 - 5 x) (2 - 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.4

Expand $(2 - 5 x) (2 - 5 x)$ using the FOIL Method.

Tap for more steps...

Step 1.4.4.1

Apply the distributive property.

$x (2 (2 - 5 x) - 5 x (2 - 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.4.2

Apply the distributive property.

$x (2 \cdot 2 + 2 (- 5 x) - 5 x (2 - 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.4.3

Apply the distributive property.

$x (2 \cdot 2 + 2 (- 5 x) - 5 x \cdot 2 - 5 x (- 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.5

Simplify and combine like terms.

Tap for more steps...

Step 1.4.5.1

Simplify each term.

Tap for more steps...

Step 1.4.5.1.1

Multiply $2$ by $2$ .

$x (4 + 2 (- 5 x) - 5 x \cdot 2 - 5 x (- 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.5.1.2

Multiply $- 5$ by $2$ .

$x (4 - 10 x - 5 x \cdot 2 - 5 x (- 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.5.1.3

Multiply $2$ by $- 5$ .

$x (4 - 10 x - 10 x - 5 x (- 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.5.1.4

Rewrite using the commutative property of multiplication.

$x (4 - 10 x - 10 x - 5 \cdot - 5 x \cdot x) (- 15 x + 2 (2 - 5 x))$

Step 1.4.5.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.4.5.1.5.1

Move $x$ .

$x (4 - 10 x - 10 x - 5 \cdot - 5 (x \cdot x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.5.1.5.2

Multiply $x$ by $x$ .

$x (4 - 10 x - 10 x - 5 \cdot - 5 x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 1.4.5.1.6

Multiply $- 5$ by $- 5$ .

$x (4 - 10 x - 10 x + 25 x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 1.4.5.2

Subtract $10 x$ from $- 10 x$ .

$x (4 - 20 x + 25 x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 1.4.6

Apply the distributive property.

$(x \cdot 4 + x (- 20 x) + x (25 x^{2})) (- 15 x + 2 (2 - 5 x))$

Step 1.4.7

Simplify.

Tap for more steps...

Step 1.4.7.1

Move $4$ to the left of $x$ .

$(4 \cdot x + x (- 20 x) + x (25 x^{2})) (- 15 x + 2 (2 - 5 x))$

Step 1.4.7.2

Rewrite using the commutative property of multiplication.

$(4 \cdot x - 20 x \cdot x + x (25 x^{2})) (- 15 x + 2 (2 - 5 x))$

Step 1.4.7.3

Rewrite using the commutative property of multiplication.

$(4 \cdot x - 20 x \cdot x + 25 x \cdot x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 1.4.8

Simplify each term.

Tap for more steps...

Step 1.4.8.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.4.8.1.1

Move $x$ .

$(4 x - 20 (x \cdot x) + 25 x \cdot x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 1.4.8.1.2

Multiply $x$ by $x$ .

$(4 x - 20 x^{2} + 25 x \cdot x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 1.4.8.2

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.4.8.2.1

Move $x^{2}$ .

$(4 x - 20 x^{2} + 25 (x^{2} x)) (- 15 x + 2 (2 - 5 x))$

Step 1.4.8.2.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 1.4.8.2.2.1

Raise $x$ to the power of $1$ .

$(4 x - 20 x^{2} + 25 (x^{2} x^{1})) (- 15 x + 2 (2 - 5 x))$

Step 1.4.8.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(4 x - 20 x^{2} + 25 x^{2 + 1}) (- 15 x + 2 (2 - 5 x))$

Step 1.4.8.2.3

Add $2$ and $1$ .

$(4 x - 20 x^{2} + 25 x^{3}) (- 15 x + 2 (2 - 5 x))$

Step 1.4.9

Simplify each term.

Tap for more steps...

Step 1.4.9.1

Apply the distributive property.

$(4 x - 20 x^{2} + 25 x^{3}) (- 15 x + 2 \cdot 2 + 2 (- 5 x))$

Step 1.4.9.2

Multiply $2$ by $2$ .

$(4 x - 20 x^{2} + 25 x^{3}) (- 15 x + 4 + 2 (- 5 x))$

Step 1.4.9.3

Multiply $- 5$ by $2$ .

$(4 x - 20 x^{2} + 25 x^{3}) (- 15 x + 4 - 10 x)$

Step 1.4.10

Subtract $10 x$ from $- 15 x$ .

$(4 x - 20 x^{2} + 25 x^{3}) (- 25 x + 4)$

Step 1.4.11

Expand $(4 x - 20 x^{2} + 25 x^{3}) (- 25 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$4 x (- 25 x) + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12

Simplify each term.

Tap for more steps...

Step 1.4.12.1

Rewrite using the commutative property of multiplication.

$4 \cdot - 25 x \cdot x + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.4.12.2.1

Move $x$ .

$4 \cdot - 25 (x \cdot x) + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.2.2

Multiply $x$ by $x$ .

$4 \cdot - 25 x^{2} + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.3

Multiply $4$ by $- 25$ .

$- 100 x^{2} + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.4

Multiply $4$ by $4$ .

$- 100 x^{2} + 16 x - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.5

Rewrite using the commutative property of multiplication.

$- 100 x^{2} + 16 x - 20 \cdot - 25 x^{2} x - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.6

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.4.12.6.1

Move $x$ .

$- 100 x^{2} + 16 x - 20 \cdot - 25 (x \cdot x^{2}) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.6.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.4.12.6.2.1

Raise $x$ to the power of $1$ .

$- 100 x^{2} + 16 x - 20 \cdot - 25 (x^{1} x^{2}) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 100 x^{2} + 16 x - 20 \cdot - 25 x^{1 + 2} - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.6.3

Add $1$ and $2$ .

$- 100 x^{2} + 16 x - 20 \cdot - 25 x^{3} - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.7

Multiply $- 20$ by $- 25$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.8

Multiply $4$ by $- 20$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 1.4.12.9

Rewrite using the commutative property of multiplication.

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 x^{3} x + 25 x^{3} \cdot 4$

Step 1.4.12.10

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.4.12.10.1

Move $x$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 (x \cdot x^{3}) + 25 x^{3} \cdot 4$

Step 1.4.12.10.2

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 1.4.12.10.2.1

Raise $x$ to the power of $1$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 (x^{1} x^{3}) + 25 x^{3} \cdot 4$

Step 1.4.12.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 x^{1 + 3} + 25 x^{3} \cdot 4$

Step 1.4.12.10.3

Add $1$ and $3$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 x^{4} + 25 x^{3} \cdot 4$

Step 1.4.12.11

Multiply $25$ by $- 25$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} - 625 x^{4} + 25 x^{3} \cdot 4$

Step 1.4.12.12

Multiply $4$ by $25$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} - 625 x^{4} + 100 x^{3}$

Step 1.4.13

Subtract $80 x^{2}$ from $- 100 x^{2}$ .

$- 180 x^{2} + 16 x + 500 x^{3} - 625 x^{4} + 100 x^{3}$

Step 1.4.14

Add $500 x^{3}$ and $100 x^{3}$ .

$- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- 180 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 625 x^{4}] + \frac{d}{d x} [600 x^{3}]$ .

$f'' (x) = \frac{d}{d x} (- 180 x^{2}) + \frac{d}{d x} (16 x) + \frac{d}{d x} (- 625 x^{4}) + \frac{d}{d x} (600 x^{3})$

Step 2.2

Evaluate $\frac{d}{d x} [- 180 x^{2}]$ .

Tap for more steps...

Step 2.2.1

Since $- 180$ is constant with respect to $x$ , the derivative of $- 180 x^{2}$ with respect to $x$ is $- 180 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - 180 \frac{d}{d x} x^{2} + \frac{d}{d x} (16 x) + \frac{d}{d x} (- 625 x^{4}) + \frac{d}{d x} (600 x^{3})$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = - 180 (2 x) + \frac{d}{d x} (16 x) + \frac{d}{d x} (- 625 x^{4}) + \frac{d}{d x} (600 x^{3})$

Step 2.2.3

Multiply $2$ by $- 180$ .

$f'' (x) = - 360 x + \frac{d}{d x} (16 x) + \frac{d}{d x} (- 625 x^{4}) + \frac{d}{d x} (600 x^{3})$

Step 2.3

Evaluate $\frac{d}{d x} [16 x]$ .

Tap for more steps...

Step 2.3.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x$ with respect to $x$ is $16 \frac{d}{d x} [x]$ .

$f'' (x) = - 360 x + 16 \frac{d}{d x} (x) + \frac{d}{d x} (- 625 x^{4}) + \frac{d}{d x} (600 x^{3})$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - 360 x + 16 \cdot 1 + \frac{d}{d x} (- 625 x^{4}) + \frac{d}{d x} (600 x^{3})$

Step 2.3.3

Multiply $16$ by $1$ .

$f'' (x) = - 360 x + 16 + \frac{d}{d x} (- 625 x^{4}) + \frac{d}{d x} (600 x^{3})$

Step 2.4

Evaluate $\frac{d}{d x} [- 625 x^{4}]$ .

Tap for more steps...

Step 2.4.1

Since $- 625$ is constant with respect to $x$ , the derivative of $- 625 x^{4}$ with respect to $x$ is $- 625 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = - 360 x + 16 - 625 \frac{d}{d x} x^{4} + \frac{d}{d x} (600 x^{3})$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = - 360 x + 16 - 625 (4 x^{3}) + \frac{d}{d x} (600 x^{3})$

Step 2.4.3

Multiply $4$ by $- 625$ .

$f'' (x) = - 360 x + 16 - 2500 x^{3} + \frac{d}{d x} (600 x^{3})$

Step 2.5

Evaluate $\frac{d}{d x} [600 x^{3}]$ .

Tap for more steps...

Step 2.5.1

Since $600$ is constant with respect to $x$ , the derivative of $600 x^{3}$ with respect to $x$ is $600 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = - 360 x + 16 - 2500 x^{3} + 600 \frac{d}{d x} (x^{3})$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = - 360 x + 16 - 2500 x^{3} + 600 (3 x^{2})$

Step 2.5.3

Multiply $3$ by $600$ .

$f'' (x) = - 360 x + 16 - 2500 x^{3} + 1800 x^{2}$

Step 2.6

Reorder terms.

$f'' (x) = - 2500 x^{3} + 1800 x^{2} - 360 x + 16$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3} = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

$x^{2} \frac{d}{d x} [{(2 - 5 x)}^{3}] + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 2 - 5 x$ .

Tap for more steps...

Step 4.1.2.1

To apply the Chain Rule, set $u$ as $2 - 5 x$ .

$x^{2} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [2 - 5 x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$x^{2} (3 u^{2} \frac{d}{d x} [2 - 5 x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.2.3

Replace all occurrences of $u$ with $2 - 5 x$ .

$x^{2} (3 {(2 - 5 x)}^{2} \frac{d}{d x} [2 - 5 x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.3

Differentiate.

Tap for more steps...

Step 4.1.3.1

By the Sum Rule, the derivative of $2 - 5 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x]$ .

$x^{2} (3 {(2 - 5 x)}^{2} (\frac{d}{d x} [2] + \frac{d}{d x} [- 5 x])) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$x^{2} (3 {(2 - 5 x)}^{2} (0 + \frac{d}{d x} [- 5 x])) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.3.3

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

$x^{2} (3 {(2 - 5 x)}^{2} \frac{d}{d x} [- 5 x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$x^{2} (3 {(2 - 5 x)}^{2} (- 5 \frac{d}{d x} [x])) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.3.5

Multiply $- 5$ by $3$ .

$x^{2} (- 15 {(2 - 5 x)}^{2} \frac{d}{d x} [x]) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{2} (- 15 {(2 - 5 x)}^{2} \cdot 1) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.3.7

Multiply $- 15$ by $1$ .

$x^{2} (- 15 {(2 - 5 x)}^{2}) + {(2 - 5 x)}^{3} \frac{d}{d x} [x^{2}]$

Step 4.1.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (- 15 {(2 - 5 x)}^{2}) + {(2 - 5 x)}^{3} (2 x)$

Step 4.1.3.9

Move $2$ to the left of ${(2 - 5 x)}^{3}$ .

$x^{2} (- 15 {(2 - 5 x)}^{2}) + 2 \cdot {(2 - 5 x)}^{3} x$

Step 4.1.4

Simplify.

Tap for more steps...

Step 4.1.4.1

Factor $x {(2 - 5 x)}^{2}$ out of $x^{2} (- 15 {(2 - 5 x)}^{2}) + 2 {(2 - 5 x)}^{3} x$ .

Tap for more steps...

Step 4.1.4.1.1

Factor $x {(2 - 5 x)}^{2}$ out of $x^{2} (- 15 {(2 - 5 x)}^{2})$ .

$x {(2 - 5 x)}^{2} (x \cdot (- 15)) + 2 {(2 - 5 x)}^{3} x$

Step 4.1.4.1.2

Factor $x {(2 - 5 x)}^{2}$ out of $2 {(2 - 5 x)}^{3} x$ .

$x {(2 - 5 x)}^{2} (x \cdot (- 15)) + x {(2 - 5 x)}^{2} (2 (2 - 5 x))$

Step 4.1.4.1.3

Factor $x {(2 - 5 x)}^{2}$ out of $x {(2 - 5 x)}^{2} (x \cdot (- 15)) + x {(2 - 5 x)}^{2} (2 (2 - 5 x))$ .

$x {(2 - 5 x)}^{2} (x \cdot (- 15) + 2 (2 - 5 x))$

Step 4.1.4.2

Move $- 15$ to the left of $x$ .

$x {(2 - 5 x)}^{2} (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.3

Rewrite ${(2 - 5 x)}^{2}$ as $(2 - 5 x) (2 - 5 x)$ .

$x ((2 - 5 x) (2 - 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.4

Expand $(2 - 5 x) (2 - 5 x)$ using the FOIL Method.

Tap for more steps...

Step 4.1.4.4.1

Apply the distributive property.

$x (2 (2 - 5 x) - 5 x (2 - 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.4.2

Apply the distributive property.

$x (2 \cdot 2 + 2 (- 5 x) - 5 x (2 - 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.4.3

Apply the distributive property.

$x (2 \cdot 2 + 2 (- 5 x) - 5 x \cdot 2 - 5 x (- 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.5

Simplify and combine like terms.

Tap for more steps...

Step 4.1.4.5.1

Simplify each term.

Tap for more steps...

Step 4.1.4.5.1.1

Multiply $2$ by $2$ .

$x (4 + 2 (- 5 x) - 5 x \cdot 2 - 5 x (- 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.5.1.2

Multiply $- 5$ by $2$ .

$x (4 - 10 x - 5 x \cdot 2 - 5 x (- 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.5.1.3

Multiply $2$ by $- 5$ .

$x (4 - 10 x - 10 x - 5 x (- 5 x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.5.1.4

Rewrite using the commutative property of multiplication.

$x (4 - 10 x - 10 x - 5 \cdot - 5 x \cdot x) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.5.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.1.4.5.1.5.1

Move $x$ .

$x (4 - 10 x - 10 x - 5 \cdot - 5 (x \cdot x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.5.1.5.2

Multiply $x$ by $x$ .

$x (4 - 10 x - 10 x - 5 \cdot - 5 x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.5.1.6

Multiply $- 5$ by $- 5$ .

$x (4 - 10 x - 10 x + 25 x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.5.2

Subtract $10 x$ from $- 10 x$ .

$x (4 - 20 x + 25 x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.6

Apply the distributive property.

$(x \cdot 4 + x (- 20 x) + x (25 x^{2})) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.7

Simplify.

Tap for more steps...

Step 4.1.4.7.1

Move $4$ to the left of $x$ .

$(4 \cdot x + x (- 20 x) + x (25 x^{2})) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.7.2

Rewrite using the commutative property of multiplication.

$(4 \cdot x - 20 x \cdot x + x (25 x^{2})) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.7.3

Rewrite using the commutative property of multiplication.

$(4 \cdot x - 20 x \cdot x + 25 x \cdot x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.8

Simplify each term.

Tap for more steps...

Step 4.1.4.8.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.1.4.8.1.1

Move $x$ .

$(4 x - 20 (x \cdot x) + 25 x \cdot x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.8.1.2

Multiply $x$ by $x$ .

$(4 x - 20 x^{2} + 25 x \cdot x^{2}) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.8.2

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 4.1.4.8.2.1

Move $x^{2}$ .

$(4 x - 20 x^{2} + 25 (x^{2} x)) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.8.2.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 4.1.4.8.2.2.1

Raise $x$ to the power of $1$ .

$(4 x - 20 x^{2} + 25 (x^{2} x^{1})) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.8.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(4 x - 20 x^{2} + 25 x^{2 + 1}) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.8.2.3

Add $2$ and $1$ .

$(4 x - 20 x^{2} + 25 x^{3}) (- 15 x + 2 (2 - 5 x))$

Step 4.1.4.9

Simplify each term.

Tap for more steps...

Step 4.1.4.9.1

Apply the distributive property.

$(4 x - 20 x^{2} + 25 x^{3}) (- 15 x + 2 \cdot 2 + 2 (- 5 x))$

Step 4.1.4.9.2

Multiply $2$ by $2$ .

$(4 x - 20 x^{2} + 25 x^{3}) (- 15 x + 4 + 2 (- 5 x))$

Step 4.1.4.9.3

Multiply $- 5$ by $2$ .

$(4 x - 20 x^{2} + 25 x^{3}) (- 15 x + 4 - 10 x)$

Step 4.1.4.10

Subtract $10 x$ from $- 15 x$ .

$(4 x - 20 x^{2} + 25 x^{3}) (- 25 x + 4)$

Step 4.1.4.11

Expand $(4 x - 20 x^{2} + 25 x^{3}) (- 25 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$4 x (- 25 x) + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12

Simplify each term.

Tap for more steps...

Step 4.1.4.12.1

Rewrite using the commutative property of multiplication.

$4 \cdot - 25 x \cdot x + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.1.4.12.2.1

Move $x$ .

$4 \cdot - 25 (x \cdot x) + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.2.2

Multiply $x$ by $x$ .

$4 \cdot - 25 x^{2} + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.3

Multiply $4$ by $- 25$ .

$- 100 x^{2} + 4 x \cdot 4 - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.4

Multiply $4$ by $4$ .

$- 100 x^{2} + 16 x - 20 x^{2} (- 25 x) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.5

Rewrite using the commutative property of multiplication.

$- 100 x^{2} + 16 x - 20 \cdot - 25 x^{2} x - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.6

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.1.4.12.6.1

Move $x$ .

$- 100 x^{2} + 16 x - 20 \cdot - 25 (x \cdot x^{2}) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.6.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 4.1.4.12.6.2.1

Raise $x$ to the power of $1$ .

$- 100 x^{2} + 16 x - 20 \cdot - 25 (x^{1} x^{2}) - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 100 x^{2} + 16 x - 20 \cdot - 25 x^{1 + 2} - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.6.3

Add $1$ and $2$ .

$- 100 x^{2} + 16 x - 20 \cdot - 25 x^{3} - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.7

Multiply $- 20$ by $- 25$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 20 x^{2} \cdot 4 + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.8

Multiply $4$ by $- 20$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 x^{3} (- 25 x) + 25 x^{3} \cdot 4$

Step 4.1.4.12.9

Rewrite using the commutative property of multiplication.

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 x^{3} x + 25 x^{3} \cdot 4$

Step 4.1.4.12.10

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.1.4.12.10.1

Move $x$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 (x \cdot x^{3}) + 25 x^{3} \cdot 4$

Step 4.1.4.12.10.2

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 4.1.4.12.10.2.1

Raise $x$ to the power of $1$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 (x^{1} x^{3}) + 25 x^{3} \cdot 4$

Step 4.1.4.12.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 x^{1 + 3} + 25 x^{3} \cdot 4$

Step 4.1.4.12.10.3

Add $1$ and $3$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} + 25 \cdot - 25 x^{4} + 25 x^{3} \cdot 4$

Step 4.1.4.12.11

Multiply $25$ by $- 25$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} - 625 x^{4} + 25 x^{3} \cdot 4$

Step 4.1.4.12.12

Multiply $4$ by $25$ .

$- 100 x^{2} + 16 x + 500 x^{3} - 80 x^{2} - 625 x^{4} + 100 x^{3}$

Step 4.1.4.13

Subtract $80 x^{2}$ from $- 100 x^{2}$ .

$- 180 x^{2} + 16 x + 500 x^{3} - 625 x^{4} + 100 x^{3}$

Step 4.1.4.14

Add $500 x^{3}$ and $100 x^{3}$ .

$f' (x) = - 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$ .

$- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3} = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3} = 0$

Step 5.2

Factor the left side of the equation.

Tap for more steps...

Step 5.2.1

Factor $x$ out of $- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$ .

Tap for more steps...

Step 5.2.1.1

Factor $x$ out of $- 180 x^{2}$ .

$x (- 180 x) + 16 x - 625 x^{4} + 600 x^{3} = 0$

Step 5.2.1.2

Factor $x$ out of $16 x$ .

$x (- 180 x) + x \cdot 16 - 625 x^{4} + 600 x^{3} = 0$

Step 5.2.1.3

Factor $x$ out of $- 625 x^{4}$ .

$x (- 180 x) + x \cdot 16 + x (- 625 x^{3}) + 600 x^{3} = 0$

Step 5.2.1.4

Factor $x$ out of $600 x^{3}$ .

$x (- 180 x) + x \cdot 16 + x (- 625 x^{3}) + x (600 x^{2}) = 0$

Step 5.2.1.5

Factor $x$ out of $x (- 180 x) + x \cdot 16$ .

$x (- 180 x + 16) + x (- 625 x^{3}) + x (600 x^{2}) = 0$

Step 5.2.1.6

Factor $x$ out of $x (- 180 x + 16) + x (- 625 x^{3})$ .

$x (- 180 x + 16 - 625 x^{3}) + x (600 x^{2}) = 0$

Step 5.2.1.7

Factor $x$ out of $x (- 180 x + 16 - 625 x^{3}) + x (600 x^{2})$ .

$x (- 180 x + 16 - 625 x^{3} + 600 x^{2}) = 0$

Step 5.2.2

Reorder terms.

$x (- 625 x^{3} + 600 x^{2} - 180 x + 16) = 0$

Step 5.2.3

Factor.

Tap for more steps...

Step 5.2.3.1

Factor $- 625 x^{3} + 600 x^{2} - 180 x + 16$ using the rational roots test.

Tap for more steps...

Step 5.2.3.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

$q = \pm 1, \pm 625, \pm 5, \pm 125, \pm 25$

Step 5.2.3.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.0016, \pm 0.2, \pm 0.008, \pm 0.04, \pm 16, \pm 0.0256, \pm 3.2, \pm 0.128, \pm 0.64, \pm 2, \pm 0.0032, \pm 0.4, \pm 0.016, \pm 0.08, \pm 8, \pm 0.0128, \pm 1.6, \pm 0.064, \pm 0.32, \pm 4, \pm 0.0064, \pm 0.8, \pm 0.032, \pm 0.16$

Step 5.2.3.1.3

Substitute $0.4$ and simplify the expression. In this case, the expression is equal to $0$ so $0.4$ is a root of the polynomial.

Tap for more steps...

Step 5.2.3.1.3.1

Substitute $0.4$ into the polynomial.

$- 625 \cdot {0.4}^{3} + 600 \cdot {0.4}^{2} - 180 \cdot 0.4 + 16$

Step 5.2.3.1.3.2

Raise $0.4$ to the power of $3$ .

$- 625 \cdot 0.064 + 600 \cdot {0.4}^{2} - 180 \cdot 0.4 + 16$

Step 5.2.3.1.3.3

Multiply $- 625$ by $0.064$ .

$- 40 + 600 \cdot {0.4}^{2} - 180 \cdot 0.4 + 16$

Step 5.2.3.1.3.4

Raise $0.4$ to the power of $2$ .

$- 40 + 600 \cdot 0.16 - 180 \cdot 0.4 + 16$

Step 5.2.3.1.3.5

Multiply $600$ by $0.16$ .

$- 40 + 96 - 180 \cdot 0.4 + 16$

Step 5.2.3.1.3.6

Add $- 40$ and $96$ .

$56 - 180 \cdot 0.4 + 16$

Step 5.2.3.1.3.7

Multiply $- 180$ by $0.4$ .

$56 - 72 + 16$

Step 5.2.3.1.3.8

Subtract $72$ from $56$ .

$- 16 + 16$

Step 5.2.3.1.3.9

Add $- 16$ and $16$ .

$0$

Step 5.2.3.1.4

Since $0.4$ is a known root, divide the polynomial by $5 x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 625 x^{3} + 600 x^{2} - 180 x + 16}{5 x - 2}$

Step 5.2.3.1.5

Divide $- 625 x^{3} + 600 x^{2} - 180 x + 16$ by $5 x - 2$ .

Tap for more steps...

Step 5.2.3.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$5 x$

$2$

$625 x^{3}$

$600 x^{2}$

$180 x$

$16$

Step 5.2.3.1.5.2

Divide the highest order term in the dividend $- 625 x^{3}$ by the highest order term in divisor $5 x$ .

				-	$125 x^{2}$
	$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$

Step 5.2.3.1.5.3

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			-	$625 x^{3}$	+	$250 x^{2}$

Step 5.2.3.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 625 x^{3} + 250 x^{2}$

			-	$125 x^{2}$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$

Step 5.2.3.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$125 x^{2}$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$

Step 5.2.3.1.5.6

Pull the next terms from the original dividend down into the current dividend.

			-	$125 x^{2}$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$

Step 5.2.3.1.5.7

Divide the highest order term in the dividend $350 x^{2}$ by the highest order term in divisor $5 x$ .

			-	$125 x^{2}$	+	$70 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$

Step 5.2.3.1.5.8

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$	+	$70 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$
					+	$350 x^{2}$	-	$140 x$

Step 5.2.3.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $350 x^{2} - 140 x$

			-	$125 x^{2}$	+	$70 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$
					-	$350 x^{2}$	+	$140 x$

Step 5.2.3.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$125 x^{2}$	+	$70 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$
					-	$350 x^{2}$	+	$140 x$
							-	$40 x$

Step 5.2.3.1.5.11

Pull the next terms from the original dividend down into the current dividend.

			-	$125 x^{2}$	+	$70 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$
					-	$350 x^{2}$	+	$140 x$
							-	$40 x$	+	$16$

Step 5.2.3.1.5.12

Divide the highest order term in the dividend $- 40 x$ by the highest order term in divisor $5 x$ .

			-	$125 x^{2}$	+	$70 x$	-	$8$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$
					-	$350 x^{2}$	+	$140 x$
							-	$40 x$	+	$16$

Step 5.2.3.1.5.13

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$	+	$70 x$	-	$8$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$
					-	$350 x^{2}$	+	$140 x$
							-	$40 x$	+	$16$
							-	$40 x$	+	$16$

Step 5.2.3.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 40 x + 16$

			-	$125 x^{2}$	+	$70 x$	-	$8$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$
					-	$350 x^{2}$	+	$140 x$
							-	$40 x$	+	$16$
							+	$40 x$	-	$16$

Step 5.2.3.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$125 x^{2}$	+	$70 x$	-	$8$
$5 x$	-	$2$	-	$625 x^{3}$	+	$600 x^{2}$	-	$180 x$	+	$16$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$350 x^{2}$	-	$180 x$
					-	$350 x^{2}$	+	$140 x$
							-	$40 x$	+	$16$
							+	$40 x$	-	$16$
										$0$

Step 5.2.3.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$- 125 x^{2} + 70 x - 8$

Step 5.2.3.1.6

Write $- 625 x^{3} + 600 x^{2} - 180 x + 16$ as a set of factors.

$x ((5 x - 2) (- 125 x^{2} + 70 x - 8)) = 0$

Step 5.2.3.2

Remove unnecessary parentheses.

$x (5 x - 2) (- 125 x^{2} + 70 x - 8) = 0$

Step 5.2.4

Factor.

Tap for more steps...

Step 5.2.4.1

Factor by grouping.

Tap for more steps...

Step 5.2.4.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 125 \cdot - 8 = 1000$ and whose sum is $b = 70$ .

Tap for more steps...

Step 5.2.4.1.1.1

Factor $70$ out of $70 x$ .

$x (5 x - 2) (- 125 x^{2} + 70 (x) - 8) = 0$

Step 5.2.4.1.1.2

Rewrite $70$ as $20$ plus $50$

$x (5 x - 2) (- 125 x^{2} + (20 + 50) x - 8) = 0$

Step 5.2.4.1.1.3

Apply the distributive property.

$x (5 x - 2) (- 125 x^{2} + 20 x + 50 x - 8) = 0$

Step 5.2.4.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.2.4.1.2.1

Group the first two terms and the last two terms.

$x (5 x - 2) ((- 125 x^{2} + 20 x) + 50 x - 8) = 0$

Step 5.2.4.1.2.2

Factor out the greatest common factor (GCF) from each group.

$x (5 x - 2) (5 x (- 25 x + 4) - 2 (- 25 x + 4)) = 0$

Step 5.2.4.1.3

Factor the polynomial by factoring out the greatest common factor, $- 25 x + 4$ .

$x (5 x - 2) ((- 25 x + 4) (5 x - 2)) = 0$

Step 5.2.4.2

Remove unnecessary parentheses.

$x (5 x - 2) (- 25 x + 4) (5 x - 2) = 0$

Step 5.2.5

Combine exponents.

Tap for more steps...

Step 5.2.5.1

Raise $5 x - 2$ to the power of $1$ .

$x ((5 x - 2) (5 x - 2)) (- 25 x + 4) = 0$

Step 5.2.5.2

Raise $5 x - 2$ to the power of $1$ .

$x ((5 x - 2) (5 x - 2)) (- 25 x + 4) = 0$

Step 5.2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x {(5 x - 2)}^{1 + 1} (- 25 x + 4) = 0$

Step 5.2.5.4

Add $1$ and $1$ .

$x {(5 x - 2)}^{2} (- 25 x + 4) = 0$

Step 5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

${(5 x - 2)}^{2} = 0$

$- 25 x + 4 = 0$

Step 5.4

Set $x$ equal to $0$ .

$x = 0$

Step 5.5

Set ${(5 x - 2)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.1

Set ${(5 x - 2)}^{2}$ equal to $0$ .

${(5 x - 2)}^{2} = 0$

Step 5.5.2

Solve ${(5 x - 2)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 5.5.2.1

Set the $5 x - 2$ equal to $0$ .

$5 x - 2 = 0$

Step 5.5.2.2

Solve for $x$ .

Tap for more steps...

Step 5.5.2.2.1

Add $2$ to both sides of the equation.

$5 x = 2$

Step 5.5.2.2.2

Divide each term in $5 x = 2$ by $5$ and simplify.

Tap for more steps...

Step 5.5.2.2.2.1

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Step 5.5.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.2.2.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.5.2.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Step 5.5.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 5.6

Set $- 25 x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.6.1

Set $- 25 x + 4$ equal to $0$ .

$- 25 x + 4 = 0$

Step 5.6.2

Solve $- 25 x + 4 = 0$ for $x$ .

Tap for more steps...

Step 5.6.2.1

Subtract $4$ from both sides of the equation.

$- 25 x = - 4$

Step 5.6.2.2

Divide each term in $- 25 x = - 4$ by $- 25$ and simplify.

Tap for more steps...

Step 5.6.2.2.1

Divide each term in $- 25 x = - 4$ by $- 25$ .

$\frac{- 25 x}{- 25} = \frac{- 4}{- 25}$

Step 5.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.6.2.2.2.1

Cancel the common factor of $- 25$ .

Tap for more steps...

Step 5.6.2.2.2.1.1

Cancel the common factor.

$\frac{- 25 x}{- 25} = \frac{- 4}{- 25}$

Step 5.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 25}$

Step 5.6.2.2.3

Simplify the right side.

Tap for more steps...

Step 5.6.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{4}{25}$

Step 5.7

The final solution is all the values that make $x {(5 x - 2)}^{2} (- 25 x + 4) = 0$ true.

$x = 0, \frac{2}{5}, \frac{4}{25}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0, \frac{2}{5}, \frac{4}{25}$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2500 {(0)}^{3} + 1800 {(0)}^{2} - 360 \cdot 0 + 16$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Raising $0$ to any positive power yields $0$ .

$- 2500 \cdot 0 + 1800 {(0)}^{2} - 360 \cdot 0 + 16$

Step 9.1.2

Multiply $- 2500$ by $0$ .

$0 + 1800 {(0)}^{2} - 360 \cdot 0 + 16$

Step 9.1.3

Raising $0$ to any positive power yields $0$ .

$0 + 1800 \cdot 0 - 360 \cdot 0 + 16$

Step 9.1.4

Multiply $1800$ by $0$ .

$0 + 0 - 360 \cdot 0 + 16$

Step 9.1.5

Multiply $- 360$ by $0$ .

$0 + 0 + 0 + 16$

Step 9.2

Simplify by adding numbers.

Tap for more steps...

Step 9.2.1

Add $0$ and $0$ .

$0 + 0 + 16$

Step 9.2.2

Add $0$ and $0$ .

$0 + 16$

Step 9.2.3

Add $0$ and $16$ .

$16$

Step 10

$x = 0$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0$ is a local minimum

Step 11

Find the y-value when $x = 0$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2} {(2 - 5 \cdot 0)}^{3}$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 {(2 - 5 \cdot 0)}^{3}$

Step 11.2.2

Multiply $- 5$ by $0$ .

$f (0) = 0 {(2 + 0)}^{3}$

Step 11.2.3

Add $2$ and $0$ .

$f (0) = 0 \cdot 2^{3}$

Step 11.2.4

Raise $2$ to the power of $3$ .

$f (0) = 0 \cdot 8$

Step 11.2.5

Multiply $0$ by $8$ .

$f (0) = 0$

Step 11.2.6

The final answer is $0$ .

$y = 0$

Step 12

Evaluate the second derivative at $x = \frac{2}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2500 {(\frac{2}{5})}^{3} + 1800 {(\frac{2}{5})}^{2} - 360 (\frac{2}{5}) + 16$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

Apply the product rule to $\frac{2}{5}$ .

$- 2500 \frac{2^{3}}{5^{3}} + 1800 {(\frac{2}{5})}^{2} - 360 (\frac{2}{5}) + 16$

Step 13.1.2

Raise $2$ to the power of $3$ .

$- 2500 \frac{8}{5^{3}} + 1800 {(\frac{2}{5})}^{2} - 360 (\frac{2}{5}) + 16$

Step 13.1.3

Raise $5$ to the power of $3$ .

$- 2500 (\frac{8}{125}) + 1800 {(\frac{2}{5})}^{2} - 360 (\frac{2}{5}) + 16$

Step 13.1.4

Cancel the common factor of $125$ .

Tap for more steps...

Step 13.1.4.1

Factor $125$ out of $- 2500$ .

$125 (- 20) \frac{8}{125} + 1800 {(\frac{2}{5})}^{2} - 360 (\frac{2}{5}) + 16$

Step 13.1.4.2

Cancel the common factor.

$125 \cdot - 20 \frac{8}{125} + 1800 {(\frac{2}{5})}^{2} - 360 (\frac{2}{5}) + 16$

Step 13.1.4.3

Rewrite the expression.

$- 20 \cdot 8 + 1800 {(\frac{2}{5})}^{2} - 360 (\frac{2}{5}) + 16$

Step 13.1.5

Multiply $- 20$ by $8$ .

$- 160 + 1800 {(\frac{2}{5})}^{2} - 360 (\frac{2}{5}) + 16$

Step 13.1.6

Apply the product rule to $\frac{2}{5}$ .

$- 160 + 1800 \frac{2^{2}}{5^{2}} - 360 (\frac{2}{5}) + 16$

Step 13.1.7

Raise $2$ to the power of $2$ .

$- 160 + 1800 \frac{4}{5^{2}} - 360 (\frac{2}{5}) + 16$

Step 13.1.8

Raise $5$ to the power of $2$ .

$- 160 + 1800 (\frac{4}{25}) - 360 (\frac{2}{5}) + 16$

Step 13.1.9

Cancel the common factor of $25$ .

Tap for more steps...

Step 13.1.9.1

Factor $25$ out of $1800$ .

$- 160 + 25 (72) \frac{4}{25} - 360 (\frac{2}{5}) + 16$

Step 13.1.9.2

Cancel the common factor.

$- 160 + 25 \cdot 72 \frac{4}{25} - 360 (\frac{2}{5}) + 16$

Step 13.1.9.3

Rewrite the expression.

$- 160 + 72 \cdot 4 - 360 (\frac{2}{5}) + 16$

Step 13.1.10

Multiply $72$ by $4$ .

$- 160 + 288 - 360 (\frac{2}{5}) + 16$

Step 13.1.11

Cancel the common factor of $5$ .

Tap for more steps...

Step 13.1.11.1

Factor $5$ out of $- 360$ .

$- 160 + 288 + 5 (- 72) \frac{2}{5} + 16$

Step 13.1.11.2

Cancel the common factor.

$- 160 + 288 + 5 \cdot - 72 \frac{2}{5} + 16$

Step 13.1.11.3

Rewrite the expression.

$- 160 + 288 - 72 \cdot 2 + 16$

Step 13.1.12

Multiply $- 72$ by $2$ .

$- 160 + 288 - 144 + 16$

Step 13.2

Simplify by adding and subtracting.

Tap for more steps...

Step 13.2.1

Add $- 160$ and $288$ .

$128 - 144 + 16$

Step 13.2.2

Subtract $144$ from $128$ .

$- 16 + 16$

Step 13.2.3

Add $- 16$ and $16$ .

$0$

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Tap for more steps...

Step 14.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, \frac{4}{25}) \cup (\frac{4}{25}, \frac{2}{5}) \cup (\frac{2}{5}, \infty)$

Step 14.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$ to check if the result is negative or positive.

Tap for more steps...

Step 14.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = - 180 {(- 2)}^{2} + 16 (- 2) - 625 {(- 2)}^{4} + 600 {(- 2)}^{3}$

Step 14.2.2

Simplify the result.

Tap for more steps...

Step 14.2.2.1

Simplify each term.

Tap for more steps...

Step 14.2.2.1.1

Raise $- 2$ to the power of $2$ .

$f' (- 2) = - 180 \cdot 4 + 16 (- 2) - 625 {(- 2)}^{4} + 600 {(- 2)}^{3}$

Step 14.2.2.1.2

Multiply $- 180$ by $4$ .

$f' (- 2) = - 720 + 16 (- 2) - 625 {(- 2)}^{4} + 600 {(- 2)}^{3}$

Step 14.2.2.1.3

Multiply $16$ by $- 2$ .

$f' (- 2) = - 720 - 32 - 625 {(- 2)}^{4} + 600 {(- 2)}^{3}$

Step 14.2.2.1.4

Raise $- 2$ to the power of $4$ .

$f' (- 2) = - 720 - 32 - 625 \cdot 16 + 600 {(- 2)}^{3}$

Step 14.2.2.1.5

Multiply $- 625$ by $16$ .

$f' (- 2) = - 720 - 32 - 10000 + 600 {(- 2)}^{3}$

Step 14.2.2.1.6

Raise $- 2$ to the power of $3$ .

$f' (- 2) = - 720 - 32 - 10000 + 600 \cdot - 8$

Step 14.2.2.1.7

Multiply $600$ by $- 8$ .

$f' (- 2) = - 720 - 32 - 10000 - 4800$

Step 14.2.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 14.2.2.2.1

Subtract $32$ from $- 720$ .

$f' (- 2) = - 752 - 10000 - 4800$

Step 14.2.2.2.2

Subtract $10000$ from $- 752$ .

$f' (- 2) = - 10752 - 4800$

Step 14.2.2.2.3

Subtract $4800$ from $- 10752$ .

$f' (- 2) = - 15552$

Step 14.2.2.3

The final answer is $- 15552$ .

$- 15552$

Step 14.3

Substitute any number, such as $0.08$ , from the interval $(0, \frac{4}{25})$ in the first derivative $- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$ to check if the result is negative or positive.

Tap for more steps...

Step 14.3.1

Replace the variable $x$ with $0.08$ in the expression.

$f' (0.08) = - 180 {(0.08)}^{2} + 16 (0.08) - 625 {(0.08)}^{4} + 600 {(0.08)}^{3}$

Step 14.3.2

Simplify the result.

Tap for more steps...

Step 14.3.2.1

Simplify each term.

Tap for more steps...

Step 14.3.2.1.1

Raise $0.08$ to the power of $2$ .

$f' (0.08) = - 180 \cdot 0.0064 + 16 (0.08) - 625 {(0.08)}^{4} + 600 {(0.08)}^{3}$

Step 14.3.2.1.2

Multiply $- 180$ by $0.0064$ .

$f' (0.08) = - 1.152 + 16 (0.08) - 625 {(0.08)}^{4} + 600 {(0.08)}^{3}$

Step 14.3.2.1.3

Multiply $16$ by $0.08$ .

$f' (0.08) = - 1.152 + 1.28 - 625 {(0.08)}^{4} + 600 {(0.08)}^{3}$

Step 14.3.2.1.4

Raise $0.08$ to the power of $4$ .

$f' (0.08) = - 1.152 + 1.28 - 625 \cdot 0.00004096 + 600 {(0.08)}^{3}$

Step 14.3.2.1.5

Multiply $- 625$ by $0.00004096$ .

$f' (0.08) = - 1.152 + 1.28 - 0.0256 + 600 {(0.08)}^{3}$

Step 14.3.2.1.6

Raise $0.08$ to the power of $3$ .

$f' (0.08) = - 1.152 + 1.28 - 0.0256 + 600 \cdot 0.000512$

Step 14.3.2.1.7

Multiply $600$ by $0.000512$ .

$f' (0.08) = - 1.152 + 1.28 - 0.0256 + 0.3072$

Step 14.3.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 14.3.2.2.1

Add $- 1.152$ and $1.28$ .

$f' (0.08) = 0.128 - 0.0256 + 0.3072$

Step 14.3.2.2.2

Subtract $0.0256$ from $0.128$ .

$f' (0.08) = 0.1024 + 0.3072$

Step 14.3.2.2.3

Add $0.1024$ and $0.3072$ .

$f' (0.08) = 0.4096$

Step 14.3.2.3

The final answer is $0.4096$ .

$0.4096$

Step 14.4

Substitute any number, such as $0.28$ , from the interval $(\frac{4}{25}, \frac{2}{5})$ in the first derivative $- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$ to check if the result is negative or positive.

Tap for more steps...

Step 14.4.1

Replace the variable $x$ with $0.28$ in the expression.

$f' (0.28) = - 180 {(0.28)}^{2} + 16 (0.28) - 625 {(0.28)}^{4} + 600 {(0.28)}^{3}$

Step 14.4.2

Simplify the result.

Tap for more steps...

Step 14.4.2.1

Simplify each term.

Tap for more steps...

Step 14.4.2.1.1

Raise $0.28$ to the power of $2$ .

$f' (0.28) = - 180 \cdot 0.0784 + 16 (0.28) - 625 {(0.28)}^{4} + 600 {(0.28)}^{3}$

Step 14.4.2.1.2

Multiply $- 180$ by $0.0784$ .

$f' (0.28) = - 14.112 + 16 (0.28) - 625 {(0.28)}^{4} + 600 {(0.28)}^{3}$

Step 14.4.2.1.3

Multiply $16$ by $0.28$ .

$f' (0.28) = - 14.112 + 4.48 - 625 {(0.28)}^{4} + 600 {(0.28)}^{3}$

Step 14.4.2.1.4

Raise $0.28$ to the power of $4$ .

$f' (0.28) = - 14.112 + 4.48 - 625 \cdot 0.00614656 + 600 {(0.28)}^{3}$

Step 14.4.2.1.5

Multiply $- 625$ by $0.00614656$ .

$f' (0.28) = - 14.112 + 4.48 - 3.8416 + 600 {(0.28)}^{3}$

Step 14.4.2.1.6

Raise $0.28$ to the power of $3$ .

$f' (0.28) = - 14.112 + 4.48 - 3.8416 + 600 \cdot 0.021952$

Step 14.4.2.1.7

Multiply $600$ by $0.021952$ .

$f' (0.28) = - 14.112 + 4.48 - 3.8416 + 13.1712$

Step 14.4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 14.4.2.2.1

Add $- 14.112$ and $4.48$ .

$f' (0.28) = - 9.632 - 3.8416 + 13.1712$

Step 14.4.2.2.2

Subtract $3.8416$ from $- 9.632$ .

$f' (0.28) = - 13.4736 + 13.1712$

Step 14.4.2.2.3

Add $- 13.4736$ and $13.1712$ .

$f' (0.28) = - 0.3024$

Step 14.4.2.3

The final answer is $- 0.3024$ .

$- 0.3024$

Step 14.5

Substitute any number, such as $3$ , from the interval $(\frac{2}{5}, \infty)$ in the first derivative $- 180 x^{2} + 16 x - 625 x^{4} + 600 x^{3}$ to check if the result is negative or positive.

Tap for more steps...

Step 14.5.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = - 180 {(3)}^{2} + 16 (3) - 625 {(3)}^{4} + 600 {(3)}^{3}$

Step 14.5.2

Simplify the result.

Tap for more steps...

Step 14.5.2.1

Simplify each term.

Tap for more steps...

Step 14.5.2.1.1

Raise $3$ to the power of $2$ .

$f' (3) = - 180 \cdot 9 + 16 (3) - 625 {(3)}^{4} + 600 {(3)}^{3}$

Step 14.5.2.1.2

Multiply $- 180$ by $9$ .

$f' (3) = - 1620 + 16 (3) - 625 {(3)}^{4} + 600 {(3)}^{3}$

Step 14.5.2.1.3

Multiply $16$ by $3$ .

$f' (3) = - 1620 + 48 - 625 {(3)}^{4} + 600 {(3)}^{3}$

Step 14.5.2.1.4

Raise $3$ to the power of $4$ .

$f' (3) = - 1620 + 48 - 625 \cdot 81 + 600 {(3)}^{3}$

Step 14.5.2.1.5

Multiply $- 625$ by $81$ .

$f' (3) = - 1620 + 48 - 50625 + 600 {(3)}^{3}$

Step 14.5.2.1.6

Raise $3$ to the power of $3$ .

$f' (3) = - 1620 + 48 - 50625 + 600 \cdot 27$

Step 14.5.2.1.7

Multiply $600$ by $27$ .

$f' (3) = - 1620 + 48 - 50625 + 16200$

Step 14.5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 14.5.2.2.1

Add $- 1620$ and $48$ .

$f' (3) = - 1572 - 50625 + 16200$

Step 14.5.2.2.2

Subtract $50625$ from $- 1572$ .

$f' (3) = - 52197 + 16200$

Step 14.5.2.2.3

Add $- 52197$ and $16200$ .

$f' (3) = - 35997$

Step 14.5.2.3

The final answer is $- 35997$ .

$- 35997$

Step 14.6

Since the first derivative changed signs from negative to positive around $x = 0$ , then $x = 0$ is a local minimum.

$x = 0$ is a local minimum

Step 14.7

Since the first derivative changed signs from positive to negative around $x = \frac{4}{25}$ , then $x = \frac{4}{25}$ is a local maximum.

$x = \frac{4}{25}$ is a local maximum

Step 14.8

Since the first derivative did not change signs around $x = \frac{2}{5}$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.9

These are the local extrema for $f (x) = x^{2} {(2 - 5 x)}^{3}$ .

$x = 0$ is a local minimum

$x = \frac{4}{25}$ is a local maximum

$x = 0$ is a local minimum

$x = \frac{4}{25}$ is a local maximum

Step 15