Calculus Examples

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

Step 1

Find the second derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

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 1.1.2

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 1.1.2.1

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 1.1.2.2

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 1.1.2.3

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 1.1.3

Differentiate.

Tap for more steps...

Step 1.1.3.1

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 1.1.3.2

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 1.1.3.3

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 1.1.3.4

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 1.1.3.5

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 1.1.3.6

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 1.1.3.7

Multiply $- 15$ by $1$ .

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

Step 1.1.3.8

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 1.1.3.9

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 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

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 1.1.4.1.1

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 1.1.4.1.2

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 1.1.4.1.3

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 1.1.4.2

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

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

Step 1.1.4.3

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 1.1.4.4

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

Tap for more steps...

Step 1.1.4.4.1

Apply the distributive property.

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

Step 1.1.4.4.2

Apply the distributive property.

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

Step 1.1.4.4.3

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 1.1.4.5

Simplify and combine like terms.

Tap for more steps...

Step 1.1.4.5.1

Simplify each term.

Tap for more steps...

Step 1.1.4.5.1.1

Multiply $2$ by $2$ .

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

Step 1.1.4.5.1.2

Multiply $- 5$ by $2$ .

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

Step 1.1.4.5.1.3

Multiply $2$ by $- 5$ .

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

Step 1.1.4.5.1.4

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 1.1.4.5.1.5

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

Tap for more steps...

Step 1.1.4.5.1.5.1

Move $x$ .

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

Step 1.1.4.5.1.5.2

Multiply $x$ by $x$ .

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

Step 1.1.4.5.1.6

Multiply $- 5$ by $- 5$ .

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

Step 1.1.4.5.2

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

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

Step 1.1.4.6

Apply the distributive property.

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

Step 1.1.4.7

Simplify.

Tap for more steps...

Step 1.1.4.7.1

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

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

Step 1.1.4.7.2

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 1.1.4.7.3

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 1.1.4.8

Simplify each term.

Tap for more steps...

Step 1.1.4.8.1

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

Tap for more steps...

Step 1.1.4.8.1.1

Move $x$ .

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

Step 1.1.4.8.1.2

Multiply $x$ by $x$ .

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

Step 1.1.4.8.2

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

Tap for more steps...

Step 1.1.4.8.2.1

Move $x^{2}$ .

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

Step 1.1.4.8.2.2

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

Tap for more steps...

Step 1.1.4.8.2.2.1

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

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

Step 1.1.4.8.2.2.2

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 1.1.4.8.2.3

Add $2$ and $1$ .

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

Step 1.1.4.9

Simplify each term.

Tap for more steps...

Step 1.1.4.9.1

Apply the distributive property.

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

Step 1.1.4.9.2

Multiply $2$ by $2$ .

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

Step 1.1.4.9.3

Multiply $- 5$ by $2$ .

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

Step 1.1.4.10

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

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

Step 1.1.4.11

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 1.1.4.12

Simplify each term.

Tap for more steps...

Step 1.1.4.12.1

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 1.1.4.12.2

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

Tap for more steps...

Step 1.1.4.12.2.1

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 1.1.4.12.2.2

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 1.1.4.12.3

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 1.1.4.12.4

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 1.1.4.12.5

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 1.1.4.12.6

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

Tap for more steps...

Step 1.1.4.12.6.1

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 1.1.4.12.6.2

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

Tap for more steps...

Step 1.1.4.12.6.2.1

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 1.1.4.12.6.2.2

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 1.1.4.12.6.3

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 1.1.4.12.7

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 1.1.4.12.8

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 1.1.4.12.9

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 1.1.4.12.10

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

Tap for more steps...

Step 1.1.4.12.10.1

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 1.1.4.12.10.2

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

Tap for more steps...

Step 1.1.4.12.10.2.1

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 1.1.4.12.10.2.2

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 1.1.4.12.10.3

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 1.1.4.12.11

Multiply $25$ by $- 25$ .

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

Step 1.1.4.12.12

Multiply $4$ by $25$ .

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

Step 1.1.4.13

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

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

Step 1.1.4.14

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

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

Step 1.2

Find the second derivative.

Tap for more steps...

Step 1.2.1

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

$\frac{d}{d x} [- 180 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 625 x^{4}] + \frac{d}{d x} [600 x^{3}]$

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

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

$- 180 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 625 x^{4}] + \frac{d}{d x} [600 x^{3}]$

Step 1.2.2.2

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

$- 180 (2 x) + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 625 x^{4}] + \frac{d}{d x} [600 x^{3}]$

Step 1.2.2.3

Multiply $2$ by $- 180$ .

$- 360 x + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 625 x^{4}] + \frac{d}{d x} [600 x^{3}]$

Step 1.2.3

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

Tap for more steps...

Step 1.2.3.1

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

$- 360 x + 16 \frac{d}{d x} [x] + \frac{d}{d x} [- 625 x^{4}] + \frac{d}{d x} [600 x^{3}]$

Step 1.2.3.2

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

$- 360 x + 16 \cdot 1 + \frac{d}{d x} [- 625 x^{4}] + \frac{d}{d x} [600 x^{3}]$

Step 1.2.3.3

Multiply $16$ by $1$ .

$- 360 x + 16 + \frac{d}{d x} [- 625 x^{4}] + \frac{d}{d x} [600 x^{3}]$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$- 360 x + 16 - 625 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [600 x^{3}]$

Step 1.2.4.2

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

$- 360 x + 16 - 625 (4 x^{3}) + \frac{d}{d x} [600 x^{3}]$

Step 1.2.4.3

Multiply $4$ by $- 625$ .

$- 360 x + 16 - 2500 x^{3} + \frac{d}{d x} [600 x^{3}]$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

$- 360 x + 16 - 2500 x^{3} + 600 \frac{d}{d x} [x^{3}]$

Step 1.2.5.2

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

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

Step 1.2.5.3

Multiply $3$ by $600$ .

$- 360 x + 16 - 2500 x^{3} + 1800 x^{2}$

Step 1.2.6

Reorder terms.

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

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- 2500 x^{3} + 1800 x^{2} - 360 x + 16$ .

$- 2500 x^{3} + 1800 x^{2} - 360 x + 16$

Step 2

Set the second derivative equal to $0$ then solve the equation $- 2500 x^{3} + 1800 x^{2} - 360 x + 16 = 0$ .

Tap for more steps...

Step 2.1

Set the second derivative equal to $0$ .

$- 2500 x^{3} + 1800 x^{2} - 360 x + 16 = 0$

Step 2.2

Factor the left side of the equation.

Tap for more steps...

Step 2.2.1

Factor $4$ out of $- 2500 x^{3} + 1800 x^{2} - 360 x + 16$ .

Tap for more steps...

Step 2.2.1.1

Factor $4$ out of $- 2500 x^{3}$ .

$4 (- 625 x^{3}) + 1800 x^{2} - 360 x + 16 = 0$

Step 2.2.1.2

Factor $4$ out of $1800 x^{2}$ .

$4 (- 625 x^{3}) + 4 (450 x^{2}) - 360 x + 16 = 0$

Step 2.2.1.3

Factor $4$ out of $- 360 x$ .

$4 (- 625 x^{3}) + 4 (450 x^{2}) + 4 (- 90 x) + 16 = 0$

Step 2.2.1.4

Factor $4$ out of $16$ .

$4 (- 625 x^{3}) + 4 (450 x^{2}) + 4 (- 90 x) + 4 (4) = 0$

Step 2.2.1.5

Factor $4$ out of $4 (- 625 x^{3}) + 4 (450 x^{2})$ .

$4 (- 625 x^{3} + 450 x^{2}) + 4 (- 90 x) + 4 (4) = 0$

Step 2.2.1.6

Factor $4$ out of $4 (- 625 x^{3} + 450 x^{2}) + 4 (- 90 x)$ .

$4 (- 625 x^{3} + 450 x^{2} - 90 x) + 4 (4) = 0$

Step 2.2.1.7

Factor $4$ out of $4 (- 625 x^{3} + 450 x^{2} - 90 x) + 4 (4)$ .

$4 (- 625 x^{3} + 450 x^{2} - 90 x + 4) = 0$

Step 2.2.2

Factor.

Tap for more steps...

Step 2.2.2.1

Factor $- 625 x^{3} + 450 x^{2} - 90 x + 4$ using the rational roots test.

Tap for more steps...

Step 2.2.2.1.1

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.

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

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

Step 2.2.2.1.2

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

$\pm 1, \pm 0.0016, \pm 0.2, \pm 0.008, \pm 0.04, \pm 4, \pm 0.0064, \pm 0.8, \pm 0.032, \pm 0.16, \pm 2, \pm 0.0032, \pm 0.4, \pm 0.016, \pm 0.08$

Step 2.2.2.1.3

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 2.2.2.1.3.1

Substitute $0.4$ into the polynomial.

$- 625 \cdot {0.4}^{3} + 450 \cdot {0.4}^{2} - 90 \cdot 0.4 + 4$

Step 2.2.2.1.3.2

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

$- 625 \cdot 0.064 + 450 \cdot {0.4}^{2} - 90 \cdot 0.4 + 4$

Step 2.2.2.1.3.3

Multiply $- 625$ by $0.064$ .

$- 40 + 450 \cdot {0.4}^{2} - 90 \cdot 0.4 + 4$

Step 2.2.2.1.3.4

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

$- 40 + 450 \cdot 0.16 - 90 \cdot 0.4 + 4$

Step 2.2.2.1.3.5

Multiply $450$ by $0.16$ .

$- 40 + 72 - 90 \cdot 0.4 + 4$

Step 2.2.2.1.3.6

Add $- 40$ and $72$ .

$32 - 90 \cdot 0.4 + 4$

Step 2.2.2.1.3.7

Multiply $- 90$ by $0.4$ .

$32 - 36 + 4$

Step 2.2.2.1.3.8

Subtract $36$ from $32$ .

$- 4 + 4$

Step 2.2.2.1.3.9

Add $- 4$ and $4$ .

$0$

Step 2.2.2.1.4

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.

$\frac{- 625 x^{3} + 450 x^{2} - 90 x + 4}{5 x - 2}$

Step 2.2.2.1.5

Divide $- 625 x^{3} + 450 x^{2} - 90 x + 4$ by $5 x - 2$ .

Tap for more steps...

Step 2.2.2.1.5.1

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

$5 x$

$2$

$625 x^{3}$

$450 x^{2}$

$90 x$

$4$

Step 2.2.2.1.5.2

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

				-	$125 x^{2}$
	$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$

Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			-	$625 x^{3}$	+	$250 x^{2}$

Step 2.2.2.1.5.4

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

			-	$125 x^{2}$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$

Step 2.2.2.1.5.5

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

			-	$125 x^{2}$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$

Step 2.2.2.1.5.6

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

			-	$125 x^{2}$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$

Step 2.2.2.1.5.7

Divide the highest order term in the dividend $200 x^{2}$ by the highest order term in divisor $5 x$ .

			-	$125 x^{2}$	+	$40 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$

Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$	+	$40 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$
					+	$200 x^{2}$	-	$80 x$

Step 2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $200 x^{2} - 80 x$

			-	$125 x^{2}$	+	$40 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$
					-	$200 x^{2}$	+	$80 x$

Step 2.2.2.1.5.10

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

			-	$125 x^{2}$	+	$40 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$
					-	$200 x^{2}$	+	$80 x$
							-	$10 x$

Step 2.2.2.1.5.11

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

			-	$125 x^{2}$	+	$40 x$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$
					-	$200 x^{2}$	+	$80 x$
							-	$10 x$	+	$4$

Step 2.2.2.1.5.12

Divide the highest order term in the dividend $- 10 x$ by the highest order term in divisor $5 x$ .

			-	$125 x^{2}$	+	$40 x$	-	$2$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$
					-	$200 x^{2}$	+	$80 x$
							-	$10 x$	+	$4$

Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

			-	$125 x^{2}$	+	$40 x$	-	$2$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$
					-	$200 x^{2}$	+	$80 x$
							-	$10 x$	+	$4$
							-	$10 x$	+	$4$

Step 2.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x + 4$

			-	$125 x^{2}$	+	$40 x$	-	$2$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$
					-	$200 x^{2}$	+	$80 x$
							-	$10 x$	+	$4$
							+	$10 x$	-	$4$

Step 2.2.2.1.5.15

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

			-	$125 x^{2}$	+	$40 x$	-	$2$
$5 x$	-	$2$	-	$625 x^{3}$	+	$450 x^{2}$	-	$90 x$	+	$4$
			+	$625 x^{3}$	-	$250 x^{2}$
					+	$200 x^{2}$	-	$90 x$
					-	$200 x^{2}$	+	$80 x$
							-	$10 x$	+	$4$
							+	$10 x$	-	$4$
										$0$

Step 2.2.2.1.5.16

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

$- 125 x^{2} + 40 x - 2$

Step 2.2.2.1.6

Write $- 625 x^{3} + 450 x^{2} - 90 x + 4$ as a set of factors.

$4 ((5 x - 2) (- 125 x^{2} + 40 x - 2)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$4 (5 x - 2) (- 125 x^{2} + 40 x - 2) = 0$

Step 2.3

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

$5 x - 2 = 0$

$- 125 x^{2} + 40 x - 2 = 0$

Step 2.4

Set $5 x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

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

$5 x - 2 = 0$

Step 2.4.2

Solve $5 x - 2 = 0$ for $x$ .

Tap for more steps...

Step 2.4.2.1

Add $2$ to both sides of the equation.

$5 x = 2$

Step 2.4.2.2

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

Tap for more steps...

Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.2.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.5

Set $- 125 x^{2} + 40 x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $- 125 x^{2} + 40 x - 2$ equal to $0$ .

$- 125 x^{2} + 40 x - 2 = 0$

Step 2.5.2

Solve $- 125 x^{2} + 40 x - 2 = 0$ for $x$ .

Tap for more steps...

Step 2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5.2.2

Substitute the values $a = - 125$ , $b = 40$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 40 \pm \sqrt{40^{2} - 4 \cdot (- 125 \cdot - 2)}}{2 \cdot - 125}$

Step 2.5.2.3

Simplify.

Tap for more steps...

Step 2.5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.5.2.3.1.1

Raise $40$ to the power of $2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot - 125 \cdot - 2}}{2 \cdot - 125}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot - 125 \cdot - 2$ .

Tap for more steps...

Step 2.5.2.3.1.2.1

Multiply $- 4$ by $- 125$ .

$x = \frac{- 40 \pm \sqrt{1600 + 500 \cdot - 2}}{2 \cdot - 125}$

Step 2.5.2.3.1.2.2

Multiply $500$ by $- 2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 1000}}{2 \cdot - 125}$

Step 2.5.2.3.1.3

Subtract $1000$ from $1600$ .

$x = \frac{- 40 \pm \sqrt{600}}{2 \cdot - 125}$

Step 2.5.2.3.1.4

Rewrite $600$ as $10^{2} \cdot 6$ .

Tap for more steps...

Step 2.5.2.3.1.4.1

Factor $100$ out of $600$ .

$x = \frac{- 40 \pm \sqrt{100 (6)}}{2 \cdot - 125}$

Step 2.5.2.3.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 40 \pm \sqrt{10^{2} \cdot 6}}{2 \cdot - 125}$

Step 2.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 40 \pm 10 \sqrt{6}}{2 \cdot - 125}$

Step 2.5.2.3.2

Multiply $2$ by $- 125$ .

$x = \frac{- 40 \pm 10 \sqrt{6}}{- 250}$

Step 2.5.2.3.3

Simplify $\frac{- 40 \pm 10 \sqrt{6}}{- 250}$ .

$x = \frac{4 \pm \sqrt{6}}{25}$

Step 2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 2.5.2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.5.2.4.1.1

Raise $40$ to the power of $2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot - 125 \cdot - 2}}{2 \cdot - 125}$

Step 2.5.2.4.1.2

Multiply $- 4 \cdot - 125 \cdot - 2$ .

Tap for more steps...

Step 2.5.2.4.1.2.1

Multiply $- 4$ by $- 125$ .

$x = \frac{- 40 \pm \sqrt{1600 + 500 \cdot - 2}}{2 \cdot - 125}$

Step 2.5.2.4.1.2.2

Multiply $500$ by $- 2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 1000}}{2 \cdot - 125}$

Step 2.5.2.4.1.3

Subtract $1000$ from $1600$ .

$x = \frac{- 40 \pm \sqrt{600}}{2 \cdot - 125}$

Step 2.5.2.4.1.4

Rewrite $600$ as $10^{2} \cdot 6$ .

Tap for more steps...

Step 2.5.2.4.1.4.1

Factor $100$ out of $600$ .

$x = \frac{- 40 \pm \sqrt{100 (6)}}{2 \cdot - 125}$

Step 2.5.2.4.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 40 \pm \sqrt{10^{2} \cdot 6}}{2 \cdot - 125}$

Step 2.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 40 \pm 10 \sqrt{6}}{2 \cdot - 125}$

Step 2.5.2.4.2

Multiply $2$ by $- 125$ .

$x = \frac{- 40 \pm 10 \sqrt{6}}{- 250}$

Step 2.5.2.4.3

Simplify $\frac{- 40 \pm 10 \sqrt{6}}{- 250}$ .

$x = \frac{4 \pm \sqrt{6}}{25}$

Step 2.5.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{4 + \sqrt{6}}{25}$

Step 2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 2.5.2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.5.2.5.1.1

Raise $40$ to the power of $2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot - 125 \cdot - 2}}{2 \cdot - 125}$

Step 2.5.2.5.1.2

Multiply $- 4 \cdot - 125 \cdot - 2$ .

Tap for more steps...

Step 2.5.2.5.1.2.1

Multiply $- 4$ by $- 125$ .

$x = \frac{- 40 \pm \sqrt{1600 + 500 \cdot - 2}}{2 \cdot - 125}$

Step 2.5.2.5.1.2.2

Multiply $500$ by $- 2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 1000}}{2 \cdot - 125}$

Step 2.5.2.5.1.3

Subtract $1000$ from $1600$ .

$x = \frac{- 40 \pm \sqrt{600}}{2 \cdot - 125}$

Step 2.5.2.5.1.4

Rewrite $600$ as $10^{2} \cdot 6$ .

Tap for more steps...

Step 2.5.2.5.1.4.1

Factor $100$ out of $600$ .

$x = \frac{- 40 \pm \sqrt{100 (6)}}{2 \cdot - 125}$

Step 2.5.2.5.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 40 \pm \sqrt{10^{2} \cdot 6}}{2 \cdot - 125}$

Step 2.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 40 \pm 10 \sqrt{6}}{2 \cdot - 125}$

Step 2.5.2.5.2

Multiply $2$ by $- 125$ .

$x = \frac{- 40 \pm 10 \sqrt{6}}{- 250}$

Step 2.5.2.5.3

Simplify $\frac{- 40 \pm 10 \sqrt{6}}{- 250}$ .

$x = \frac{4 \pm \sqrt{6}}{25}$

Step 2.5.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{4 - \sqrt{6}}{25}$

Step 2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{4 + \sqrt{6}}{25}, \frac{4 - \sqrt{6}}{25}$

Step 2.6

The final solution is all the values that make $4 (5 x - 2) (- 125 x^{2} + 40 x - 2) = 0$ true.

$x = \frac{2}{5}, \frac{4 + \sqrt{6}}{25}, \frac{4 - \sqrt{6}}{25}$

Step 3

Find the points where the second derivative is $0$ .

Tap for more steps...

Step 3.1

Substitute $\frac{2}{5}$ in $f (x) = x^{2} {(2 - 5 x)}^{3}$ to find the value of $y$ .

Tap for more steps...

Step 3.1.1

Replace the variable $x$ with $\frac{2}{5}$ in the expression.

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

Step 3.1.2

Simplify the result.

Tap for more steps...

Step 3.1.2.1

Simplify the expression.

Tap for more steps...

Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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

$f (\frac{2}{5}) = \frac{4}{5^{2}} \cdot {(2 - 5 (\frac{2}{5}))}^{3}$

Step 3.1.2.1.3

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

$f (\frac{2}{5}) = \frac{4}{25} \cdot {(2 - 5 (\frac{2}{5}))}^{3}$

Step 3.1.2.2

Simplify each term.

Tap for more steps...

Step 3.1.2.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.1.2.2.1.1

Factor $5$ out of $- 5$ .

$f (\frac{2}{5}) = \frac{4}{25} \cdot {(2 + 5 (- 1) (\frac{2}{5}))}^{3}$

Step 3.1.2.2.1.2

Cancel the common factor.

$f (\frac{2}{5}) = \frac{4}{25} \cdot {(2 + 5 \cdot (- 1 (\frac{2}{5})))}^{3}$

Step 3.1.2.2.1.3

Rewrite the expression.

$f (\frac{2}{5}) = \frac{4}{25} \cdot {(2 - 1 \cdot 2)}^{3}$

Step 3.1.2.2.2

Multiply $- 1$ by $2$ .

$f (\frac{2}{5}) = \frac{4}{25} \cdot {(2 - 2)}^{3}$

Step 3.1.2.3

Simplify the expression.

Tap for more steps...

Step 3.1.2.3.1

Subtract $2$ from $2$ .

$f (\frac{2}{5}) = \frac{4}{25} \cdot 0^{3}$

Step 3.1.2.3.2

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

$f (\frac{2}{5}) = \frac{4}{25} \cdot 0$

Step 3.1.2.3.3

Multiply $\frac{4}{25}$ by $0$ .

$f (\frac{2}{5}) = 0$

Step 3.1.2.4

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $\frac{2}{5}$ in $f (x) = x^{2} {(2 - 5 x)}^{3}$ is $(\frac{2}{5}, 0)$ . This point can be an inflection point.

$(\frac{2}{5}, 0)$

Step 3.3

Substitute $0.25797958$ in $f (x) = x^{2} {(2 - 5 x)}^{3}$ to find the value of $y$ .

Tap for more steps...

Step 3.3.1

Replace the variable $x$ with $0.25797958$ in the expression.

$f (0.25797958) = {(0.25797958)}^{2} {(2 - 5 \cdot 0.25797958)}^{3}$

Step 3.3.2

Simplify the result.

Tap for more steps...

Step 3.3.2.1

Raise $0.25797958$ to the power of $2$ .

$f (0.25797958) = 0.06655346 {(2 - 5 \cdot 0.25797958)}^{3}$

Step 3.3.2.2

Multiply $- 5$ by $0.25797958$ .

$f (0.25797958) = 0.06655346 {(2 - 1.28989794)}^{3}$

Step 3.3.2.3

Subtract $1.28989794$ from $2$ .

$f (0.25797958) = 0.06655346 \cdot {0.71010205}^{3}$

Step 3.3.2.4

Raise $0.71010205$ to the power of $3$ .

$f (0.25797958) = 0.06655346 \cdot 0.35806535$

Step 3.3.2.5

Multiply $0.06655346$ by $0.35806535$ .

$f (0.25797958) = 0.02383049$

Step 3.3.2.6

The final answer is $0.02383049$ .

$0.02383049$

Step 3.4

The point found by substituting $0.25797958$ in $f (x) = x^{2} {(2 - 5 x)}^{3}$ is $(0.25797958, 0.02383049)$ . This point can be an inflection point.

$(0.25797958, 0.02383049)$

Step 3.5

Substitute $0.06202041$ in $f (x) = x^{2} {(2 - 5 x)}^{3}$ to find the value of $y$ .

Tap for more steps...

Step 3.5.1

Replace the variable $x$ with $0.06202041$ in the expression.

$f (0.06202041) = {(0.06202041)}^{2} {(2 - 5 \cdot 0.06202041)}^{3}$

Step 3.5.2

Simplify the result.

Tap for more steps...

Step 3.5.2.1

Raise $0.06202041$ to the power of $2$ .

$f (0.06202041) = 0.00384653 {(2 - 5 \cdot 0.06202041)}^{3}$

Step 3.5.2.2

Multiply $- 5$ by $0.06202041$ .

$f (0.06202041) = 0.00384653 {(2 - 0.31010205)}^{3}$

Step 3.5.2.3

Subtract $0.31010205$ from $2$ .

$f (0.06202041) = 0.00384653 \cdot {1.68989794}^{3}$

Step 3.5.2.4

Raise $1.68989794$ to the power of $3$ .

$f (0.06202041) = 0.00384653 \cdot 4.82593464$

Step 3.5.2.5

Multiply $0.00384653$ by $4.82593464$ .

$f (0.06202041) = 0.0185631$

Step 3.5.2.6

The final answer is $0.0185631$ .

$0.0185631$

Step 3.6

The point found by substituting $0.06202041$ in $f (x) = x^{2} {(2 - 5 x)}^{3}$ is $(0.06202041, 0.0185631)$ . This point can be an inflection point.

$(0.06202041, 0.0185631)$

Step 3.7

Determine the points that could be inflection points.

$(\frac{2}{5}, 0), (0.25797958, 0.02383049), (0.06202041, 0.0185631)$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, 0.06202041) \cup (0.06202041, 0.25797958) \cup (0.25797958, \frac{2}{5}) \cup (\frac{2}{5}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0.06202041)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 0.03797958$ in the expression.

$f'' (- 0.03797958) = - 2500 {(- 0.03797958)}^{3} + 1800 {(- 0.03797958)}^{2} - 360 \cdot - 0.03797958 + 16$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Raise $- 0.03797958$ to the power of $3$ .

$f'' (- 0.03797958) = - 2500 \cdot - 0.00005478 + 1800 {(- 0.03797958)}^{2} - 360 \cdot - 0.03797958 + 16$

Step 5.2.1.2

Multiply $- 2500$ by $- 0.00005478$ .

$f'' (- 0.03797958) = 0.13695907 + 1800 {(- 0.03797958)}^{2} - 360 \cdot - 0.03797958 + 16$

Step 5.2.1.3

Raise $- 0.03797958$ to the power of $2$ .

$f'' (- 0.03797958) = 0.13695907 + 1800 \cdot 0.00144244 - 360 \cdot - 0.03797958 + 16$

Step 5.2.1.4

Multiply $1800$ by $0.00144244$ .

$f'' (- 0.03797958) = 0.13695907 + 2.59640862 - 360 \cdot - 0.03797958 + 16$

Step 5.2.1.5

Multiply $- 360$ by $- 0.03797958$ .

$f'' (- 0.03797958) = 0.13695907 + 2.59640862 + 13.67265229 + 16$

Step 5.2.2

Simplify by adding numbers.

Tap for more steps...

Step 5.2.2.1

Add $0.13695907$ and $2.59640862$ .

$f'' (- 0.03797958) = 2.73336769 + 13.67265229 + 16$

Step 5.2.2.2

Add $2.73336769$ and $13.67265229$ .

$f'' (- 0.03797958) = 16.40601999 + 16$

Step 5.2.2.3

Add $16.40601999$ and $16$ .

$f'' (- 0.03797958) = 32.40601999$

Step 5.2.3

The final answer is $32.40601999$ .

$32.40601999$

Step 5.3

At $- 0.03797958$ , the second derivative is $32.40601999$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0.06202041)$ .

Increasing on $(- \infty, 0.06202041)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(0.06202041, 0.25797958)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $0.15\overline{9}$ in the expression.

$f'' (0.15\overline{9}) = - 2500 {(0.15\overline{9})}^{3} + 1800 {(0.15\overline{9})}^{2} - 360 \cdot 0.15\overline{9} + 16$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Raise $0.15\overline{9}$ to the power of $3$ .

$f'' (0.15\overline{9}) = - 2500 \cdot 0.004096 + 1800 {(0.15\overline{9})}^{2} - 360 \cdot 0.15\overline{9} + 16$

Step 6.2.1.2

Multiply $- 2500$ by $0.004096$ .

$f'' (0.15\overline{9}) = - 10.24 + 1800 {(0.15\overline{9})}^{2} - 360 \cdot 0.15\overline{9} + 16$

Step 6.2.1.3

Raise $0.15\overline{9}$ to the power of $2$ .

$f'' (0.15\overline{9}) = - 10.24 + 1800 \cdot 0.0256 - 360 \cdot 0.15\overline{9} + 16$

Step 6.2.1.4

Multiply $1800$ by $0.0256$ .

$f'' (0.15\overline{9}) = - 10.24 + 46.08 - 360 \cdot 0.15\overline{9} + 16$

Step 6.2.1.5

Multiply $- 360$ by $0.15\overline{9}$ .

$f'' (0.15\overline{9}) = - 10.24 + 46.08 - 57.6 + 16$

Step 6.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 6.2.2.1

Add $- 10.24$ and $46.08$ .

$f'' (0.15\overline{9}) = 35.84 - 57.6 + 16$

Step 6.2.2.2

Subtract $57.6$ from $35.84$ .

$f'' (0.15\overline{9}) = - 21.76 + 16$

Step 6.2.2.3

Add $- 21.76$ and $16$ .

$f'' (0.15\overline{9}) = - 5.76$

Step 6.2.3

The final answer is $- 5.76$ .

$- 5.76$

Step 6.3

At $0.15\overline{9}$ , the second derivative is $- 5.76$ . Since this is negative, the second derivative is decreasing on the interval $(0.06202041, 0.25797958)$

Decreasing on $(0.06202041, 0.25797958)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0.25797958, \frac{2}{5})$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $0.32898979$ in the expression.

$f'' (0.32898979) = - 2500 {(0.32898979)}^{3} + 1800 {(0.32898979)}^{2} - 360 \cdot 0.32898979 + 16$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

Raise $0.32898979$ to the power of $3$ .

$f'' (0.32898979) = - 2500 \cdot 0.03560797 + 1800 {(0.32898979)}^{2} - 360 \cdot 0.32898979 + 16$

Step 7.2.1.2

Multiply $- 2500$ by $0.03560797$ .

$f'' (0.32898979) = - 89.01993814 + 1800 {(0.32898979)}^{2} - 360 \cdot 0.32898979 + 16$

Step 7.2.1.3

Raise $0.32898979$ to the power of $2$ .

$f'' (0.32898979) = - 89.01993814 + 1800 \cdot 0.10823428 - 360 \cdot 0.32898979 + 16$

Step 7.2.1.4

Multiply $1800$ by $0.10823428$ .

$f'' (0.32898979) = - 89.01993814 + 194.82171321 - 360 \cdot 0.32898979 + 16$

Step 7.2.1.5

Multiply $- 360$ by $0.32898979$ .

$f'' (0.32898979) = - 89.01993814 + 194.82171321 - 118.43632614 + 16$

Step 7.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 7.2.2.1

Add $- 89.01993814$ and $194.82171321$ .

$f'' (0.32898979) = 105.80177507 - 118.43632614 + 16$

Step 7.2.2.2

Subtract $118.43632614$ from $105.80177507$ .

$f'' (0.32898979) = - 12.63455107 + 16$

Step 7.2.2.3

Add $- 12.63455107$ and $16$ .

$f'' (0.32898979) = 3.36544892$

Step 7.2.3

The final answer is $3.36544892$ .

$3.36544892$

Step 7.3

At $0.32898979$ , the second derivative is $3.36544892$ . Since this is positive, the second derivative is increasing on the interval $(0.25797958, \frac{2}{5})$ .

Increasing on $(0.25797958, \frac{2}{5})$ since $f'' (x) > 0$

Step 8

Substitute a value from the interval $(\frac{2}{5}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $0.5$ in the expression.

$f'' (0.5) = - 2500 {(0.5)}^{3} + 1800 {(0.5)}^{2} - 360 \cdot 0.5 + 16$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify each term.

Tap for more steps...

Step 8.2.1.1

Raise $0.5$ to the power of $3$ .

$f'' (0.5) = - 2500 \cdot 0.125 + 1800 {(0.5)}^{2} - 360 \cdot 0.5 + 16$

Step 8.2.1.2

Multiply $- 2500$ by $0.125$ .

$f'' (0.5) = - 312.5 + 1800 {(0.5)}^{2} - 360 \cdot 0.5 + 16$

Step 8.2.1.3

Raise $0.5$ to the power of $2$ .

$f'' (0.5) = - 312.5 + 1800 \cdot 0.25 - 360 \cdot 0.5 + 16$

Step 8.2.1.4

Multiply $1800$ by $0.25$ .

$f'' (0.5) = - 312.5 + 450 - 360 \cdot 0.5 + 16$

Step 8.2.1.5

Multiply $- 360$ by $0.5$ .

$f'' (0.5) = - 312.5 + 450 - 180 + 16$

Step 8.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 8.2.2.1

Add $- 312.5$ and $450$ .

$f'' (0.5) = 137.5 - 180 + 16$

Step 8.2.2.2

Subtract $180$ from $137.5$ .

$f'' (0.5) = - 42.5 + 16$

Step 8.2.2.3

Add $- 42.5$ and $16$ .

$f'' (0.5) = - 26.5$

Step 8.2.3

The final answer is $- 26.5$ .

$- 26.5$

Step 8.3

At $0.5$ , the second derivative is $- 26.5$ . Since this is negative, the second derivative is decreasing on the interval $(\frac{2}{5}, \infty)$

Decreasing on $(\frac{2}{5}, \infty)$ since $f'' (x) < 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(0.06202041, 0.0185631), (0.25797958, 0.02383049), (\frac{2}{5}, 0)$ .

$(0.06202041, 0.0185631), (0.25797958, 0.02383049), (\frac{2}{5}, 0)$

Step 10