Calculus Examples

$f (x) = x^{2} {(4 - 2 x)}^{2}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Rewrite ${(4 - 2 x)}^{2}$ as $(4 - 2 x) (4 - 2 x)$ .

$\frac{d}{d x} [x^{2} ((4 - 2 x) (4 - 2 x))]$

Step 1.1.2

Expand $(4 - 2 x) (4 - 2 x)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$\frac{d}{d x} [x^{2} (4 (4 - 2 x) - 2 x (4 - 2 x))]$

Step 1.1.2.2

Apply the distributive property.

$\frac{d}{d x} [x^{2} (4 \cdot 4 + 4 (- 2 x) - 2 x (4 - 2 x))]$

Step 1.1.2.3

Apply the distributive property.

$\frac{d}{d x} [x^{2} (4 \cdot 4 + 4 (- 2 x) - 2 x \cdot 4 - 2 x (- 2 x))]$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $4$ by $4$ .

$\frac{d}{d x} [x^{2} (16 + 4 (- 2 x) - 2 x \cdot 4 - 2 x (- 2 x))]$

Step 1.1.3.1.2

Multiply $- 2$ by $4$ .

$\frac{d}{d x} [x^{2} (16 - 8 x - 2 x \cdot 4 - 2 x (- 2 x))]$

Step 1.1.3.1.3

Multiply $4$ by $- 2$ .

$\frac{d}{d x} [x^{2} (16 - 8 x - 8 x - 2 x (- 2 x))]$

Step 1.1.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [x^{2} (16 - 8 x - 8 x - 2 \cdot - 2 x \cdot x)]$

Step 1.1.3.1.5

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

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Step 1.1.3.1.5.1

Move $x$ .

$\frac{d}{d x} [x^{2} (16 - 8 x - 8 x - 2 \cdot - 2 (x \cdot x))]$

Step 1.1.3.1.5.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{2} (16 - 8 x - 8 x - 2 \cdot - 2 x^{2})]$

Step 1.1.3.1.6

Multiply $- 2$ by $- 2$ .

$\frac{d}{d x} [x^{2} (16 - 8 x - 8 x + 4 x^{2})]$

Step 1.1.3.2

Subtract $8 x$ from $- 8 x$ .

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

Step 1.1.4

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 $g (x) = 16 - 16 x + 4 x^{2}$ .

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

Step 1.1.5

Differentiate.

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Step 1.1.5.1

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

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

Step 1.1.5.2

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$x^{2} (0 + \frac{d}{d x} [- 16 x] + \frac{d}{d x} [4 x^{2}]) + (16 - 16 x + 4 x^{2}) \frac{d}{d x} [x^{2}]$

Step 1.1.5.3

Add $0$ and $\frac{d}{d x} [- 16 x]$ .

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

Step 1.1.5.4

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

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

Step 1.1.5.5

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

$x^{2} (- 16 \cdot 1 + \frac{d}{d x} [4 x^{2}]) + (16 - 16 x + 4 x^{2}) \frac{d}{d x} [x^{2}]$

Step 1.1.5.6

Multiply $- 16$ by $1$ .

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

Step 1.1.5.7

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

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

Step 1.1.5.8

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

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

Step 1.1.5.9

Multiply $2$ by $4$ .

$x^{2} (- 16 + 8 x) + (16 - 16 x + 4 x^{2}) \frac{d}{d x} [x^{2}]$

Step 1.1.5.10

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

$x^{2} (- 16 + 8 x) + (16 - 16 x + 4 x^{2}) (2 x)$

Step 1.1.5.11

Move $2$ to the left of $16 - 16 x + 4 x^{2}$ .

$x^{2} (- 16 + 8 x) + 2 \cdot (16 - 16 x + 4 x^{2}) x$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

$x^{2} \cdot - 16 + x^{2} (8 x) + 2 (16 - 16 x + 4 x^{2}) x$

Step 1.1.6.2

Apply the distributive property.

$x^{2} \cdot - 16 + x^{2} (8 x) + (2 \cdot 16 + 2 (- 16 x) + 2 (4 x^{2})) x$

Step 1.1.6.3

Apply the distributive property.

$x^{2} \cdot - 16 + x^{2} (8 x) + 2 \cdot 16 x + 2 (- 16 x) x + 2 (4 x^{2}) x$

Step 1.1.6.4

Combine terms.

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Step 1.1.6.4.1

Move $- 16$ to the left of $x^{2}$ .

$- 16 \cdot x^{2} + x^{2} (8 x) + 2 \cdot 16 x + 2 (- 16 x) x + 2 (4 x^{2}) x$

Step 1.1.6.4.2

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

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Step 1.1.6.4.2.1

Move $x$ .

$- 16 x^{2} + x \cdot x^{2} \cdot 8 + 2 \cdot 16 x + 2 (- 16 x) x + 2 (4 x^{2}) x$

Step 1.1.6.4.2.2

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

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Step 1.1.6.4.2.2.1

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

$- 16 x^{2} + x^{1} x^{2} \cdot 8 + 2 \cdot 16 x + 2 (- 16 x) x + 2 (4 x^{2}) x$

Step 1.1.6.4.2.2.2

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

$- 16 x^{2} + x^{1 + 2} \cdot 8 + 2 \cdot 16 x + 2 (- 16 x) x + 2 (4 x^{2}) x$

Step 1.1.6.4.2.3

Add $1$ and $2$ .

$- 16 x^{2} + x^{3} \cdot 8 + 2 \cdot 16 x + 2 (- 16 x) x + 2 (4 x^{2}) x$

Step 1.1.6.4.3

Move $8$ to the left of $x^{3}$ .

$- 16 x^{2} + 8 \cdot x^{3} + 2 \cdot 16 x + 2 (- 16 x) x + 2 (4 x^{2}) x$

Step 1.1.6.4.4

Multiply $2$ by $16$ .

$- 16 x^{2} + 8 x^{3} + 32 x + 2 (- 16 x) x + 2 (4 x^{2}) x$

Step 1.1.6.4.5

Multiply $- 16$ by $2$ .

$- 16 x^{2} + 8 x^{3} + 32 x - 32 x \cdot x + 2 (4 x^{2}) x$

Step 1.1.6.4.6

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

$- 16 x^{2} + 8 x^{3} + 32 x - 32 (x^{1} x) + 2 (4 x^{2}) x$

Step 1.1.6.4.7

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

$- 16 x^{2} + 8 x^{3} + 32 x - 32 (x^{1} x^{1}) + 2 (4 x^{2}) x$

Step 1.1.6.4.8

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

$- 16 x^{2} + 8 x^{3} + 32 x - 32 x^{1 + 1} + 2 (4 x^{2}) x$

Step 1.1.6.4.9

Add $1$ and $1$ .

$- 16 x^{2} + 8 x^{3} + 32 x - 32 x^{2} + 2 (4 x^{2}) x$

Step 1.1.6.4.10

Multiply $4$ by $2$ .

$- 16 x^{2} + 8 x^{3} + 32 x - 32 x^{2} + 8 x^{2} x$

Step 1.1.6.4.11

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

$- 16 x^{2} + 8 x^{3} + 32 x - 32 x^{2} + 8 (x^{1} x^{2})$

Step 1.1.6.4.12

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

$- 16 x^{2} + 8 x^{3} + 32 x - 32 x^{2} + 8 x^{1 + 2}$

Step 1.1.6.4.13

Add $1$ and $2$ .

$- 16 x^{2} + 8 x^{3} + 32 x - 32 x^{2} + 8 x^{3}$

Step 1.1.6.4.14

Subtract $32 x^{2}$ from $- 16 x^{2}$ .

$- 48 x^{2} + 8 x^{3} + 32 x + 8 x^{3}$

Step 1.1.6.4.15

Add $8 x^{3}$ and $8 x^{3}$ .

$- 48 x^{2} + 16 x^{3} + 32 x$

Step 1.1.6.5

Reorder terms.

$f' (x) = 16 x^{3} - 48 x^{2} + 32 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $16 x^{3} - 48 x^{2} + 32 x$ with respect to $x$ is $\frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [32 x]$ .

$\frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [32 x]$

Step 1.2.2

Evaluate $\frac{d}{d x} [16 x^{3}]$ .

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Step 1.2.2.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{3}$ with respect to $x$ is $16 \frac{d}{d x} [x^{3}]$ .

$16 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [32 x]$

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 $n = 3$ .

$16 (3 x^{2}) + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [32 x]$

Step 1.2.2.3

Multiply $3$ by $16$ .

$48 x^{2} + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [32 x]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 48 x^{2}]$ .

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Step 1.2.3.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x^{2}$ with respect to $x$ is $- 48 \frac{d}{d x} [x^{2}]$ .

$48 x^{2} - 48 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [32 x]$

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 $n = 2$ .

$48 x^{2} - 48 (2 x) + \frac{d}{d x} [32 x]$

Step 1.2.3.3

Multiply $2$ by $- 48$ .

$48 x^{2} - 96 x + \frac{d}{d x} [32 x]$

Step 1.2.4

Evaluate $\frac{d}{d x} [32 x]$ .

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Step 1.2.4.1

Since $32$ is constant with respect to $x$ , the derivative of $32 x$ with respect to $x$ is $32 \frac{d}{d x} [x]$ .

$48 x^{2} - 96 x + 32 \frac{d}{d x} [x]$

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 $n = 1$ .

$48 x^{2} - 96 x + 32 \cdot 1$

Step 1.2.4.3

Multiply $32$ by $1$ .

$f'' (x) = 48 x^{2} - 96 x + 32$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $48 x^{2} - 96 x + 32$ .

$48 x^{2} - 96 x + 32$

Step 2

Set the second derivative equal to $0$ then solve the equation $48 x^{2} - 96 x + 32 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$48 x^{2} - 96 x + 32 = 0$

Step 2.2

Factor $16$ out of $48 x^{2} - 96 x + 32$ .

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Step 2.2.1

Factor $16$ out of $48 x^{2}$ .

$16 (3 x^{2}) - 96 x + 32 = 0$

Step 2.2.2

Factor $16$ out of $- 96 x$ .

$16 (3 x^{2}) + 16 (- 6 x) + 32 = 0$

Step 2.2.3

Factor $16$ out of $32$ .

$16 (3 x^{2}) + 16 (- 6 x) + 16 (2) = 0$

Step 2.2.4

Factor $16$ out of $16 (3 x^{2}) + 16 (- 6 x)$ .

$16 (3 x^{2} - 6 x) + 16 (2) = 0$

Step 2.2.5

Factor $16$ out of $16 (3 x^{2} - 6 x) + 16 (2)$ .

$16 (3 x^{2} - 6 x + 2) = 0$

Step 2.3

Divide each term in $16 (3 x^{2} - 6 x + 2) = 0$ by $16$ and simplify.

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Step 2.3.1

Divide each term in $16 (3 x^{2} - 6 x + 2) = 0$ by $16$ .

$\frac{16 (3 x^{2} - 6 x + 2)}{16} = \frac{0}{16}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $16$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{16 (3 x^{2} - 6 x + 2)}{16} = \frac{0}{16}$

Step 2.3.2.1.2

Divide $3 x^{2} - 6 x + 2$ by $1$ .

$3 x^{2} - 6 x + 2 = \frac{0}{16}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $16$ .

$3 x^{2} - 6 x + 2 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

Substitute the values $a = 3$ , $b = - 6$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (3 \cdot 2)}}{2 \cdot 3}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot 2}}{2 \cdot 3}$

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot 2$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{6 \pm \sqrt{36 - 12 \cdot 2}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply $- 12$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 3}$

Step 2.6.1.3

Subtract $24$ from $36$ .

$x = \frac{6 \pm \sqrt{12}}{2 \cdot 3}$

Step 2.6.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.6.1.4.1

Factor $4$ out of $12$ .

$x = \frac{6 \pm \sqrt{4 (3)}}{2 \cdot 3}$

Step 2.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3}}{2 \cdot 3}$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{6 \pm 2 \sqrt{3}}{6}$

Step 2.6.3

Simplify $\frac{6 \pm 2 \sqrt{3}}{6}$ .

$x = \frac{3 \pm \sqrt{3}}{3}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot 2}}{2 \cdot 3}$

Step 2.7.1.2

Multiply $- 4 \cdot 3 \cdot 2$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{6 \pm \sqrt{36 - 12 \cdot 2}}{2 \cdot 3}$

Step 2.7.1.2.2

Multiply $- 12$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 3}$

Step 2.7.1.3

Subtract $24$ from $36$ .

$x = \frac{6 \pm \sqrt{12}}{2 \cdot 3}$

Step 2.7.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.7.1.4.1

Factor $4$ out of $12$ .

$x = \frac{6 \pm \sqrt{4 (3)}}{2 \cdot 3}$

Step 2.7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 3}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3}}{2 \cdot 3}$

Step 2.7.2

Multiply $2$ by $3$ .

$x = \frac{6 \pm 2 \sqrt{3}}{6}$

Step 2.7.3

Simplify $\frac{6 \pm 2 \sqrt{3}}{6}$ .

$x = \frac{3 \pm \sqrt{3}}{3}$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{3}}{3}$

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot 2}}{2 \cdot 3}$

Step 2.8.1.2

Multiply $- 4 \cdot 3 \cdot 2$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{6 \pm \sqrt{36 - 12 \cdot 2}}{2 \cdot 3}$

Step 2.8.1.2.2

Multiply $- 12$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 3}$

Step 2.8.1.3

Subtract $24$ from $36$ .

$x = \frac{6 \pm \sqrt{12}}{2 \cdot 3}$

Step 2.8.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.8.1.4.1

Factor $4$ out of $12$ .

$x = \frac{6 \pm \sqrt{4 (3)}}{2 \cdot 3}$

Step 2.8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 3}$

Step 2.8.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3}}{2 \cdot 3}$

Step 2.8.2

Multiply $2$ by $3$ .

$x = \frac{6 \pm 2 \sqrt{3}}{6}$

Step 2.8.3

Simplify $\frac{6 \pm 2 \sqrt{3}}{6}$ .

$x = \frac{3 \pm \sqrt{3}}{3}$

Step 2.8.4

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{3}}{3}$

Step 2.9

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{3}}{3}, \frac{3 - \sqrt{3}}{3}$

Step 3

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

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Step 3.1

Substitute $1.57735026$ in $f (x) = x^{2} {(4 - 2 x)}^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $1.57735026$ in the expression.

$f (1.57735026) = {(1.57735026)}^{2} {(4 - 2 \cdot 1.57735026)}^{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Raise $1.57735026$ to the power of $2$ .

$f (1.57735026) = 2.48803387 {(4 - 2 \cdot 1.57735026)}^{2}$

Step 3.1.2.2

Multiply $- 2$ by $1.57735026$ .

$f (1.57735026) = 2.48803387 {(4 - 3.15470053)}^{2}$

Step 3.1.2.3

Subtract $3.15470053$ from $4$ .

$f (1.57735026) = 2.48803387 \cdot {0.84529946}^{2}$

Step 3.1.2.4

Raise $0.84529946$ to the power of $2$ .

$f (1.57735026) = 2.48803387 \cdot 0.71453117$

Step 3.1.2.5

Multiply $2.48803387$ by $0.71453117$ .

$f (1.57735026) = 1.77777777$

Step 3.1.2.6

The final answer is $1.77777777$ .

$1.77777777$

Step 3.2

The point found by substituting $1.57735026$ in $f (x) = x^{2} {(4 - 2 x)}^{2}$ is $(1.57735026, 1.77777777)$ . This point can be an inflection point.

$(1.57735026, 1.77777777)$

Step 3.3

Substitute $0.42264973$ in $f (x) = x^{2} {(4 - 2 x)}^{2}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $0.42264973$ in the expression.

$f (0.42264973) = {(0.42264973)}^{2} {(4 - 2 \cdot 0.42264973)}^{2}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Raise $0.42264973$ to the power of $2$ .

$f (0.42264973) = 0.17863279 {(4 - 2 \cdot 0.42264973)}^{2}$

Step 3.3.2.2

Multiply $- 2$ by $0.42264973$ .

$f (0.42264973) = 0.17863279 {(4 - 0.84529946)}^{2}$

Step 3.3.2.3

Subtract $0.84529946$ from $4$ .

$f (0.42264973) = 0.17863279 \cdot {3.15470053}^{2}$

Step 3.3.2.4

Raise $3.15470053$ to the power of $2$ .

$f (0.42264973) = 0.17863279 \cdot 9.95213548$

Step 3.3.2.5

Multiply $0.17863279$ by $9.95213548$ .

$f (0.42264973) = 1.77777777$

Step 3.3.2.6

The final answer is $1.77777777$ .

$1.77777777$

Step 3.4

The point found by substituting $0.42264973$ in $f (x) = x^{2} {(4 - 2 x)}^{2}$ is $(0.42264973, 1.77777777)$ . This point can be an inflection point.

$(0.42264973, 1.77777777)$

Step 3.5

Determine the points that could be inflection points.

$(1.57735026, 1.77777777), (0.42264973, 1.77777777)$

Step 4

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

$(- \infty, 0.42264973) \cup (0.42264973, 1.57735026) \cup (1.57735026, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0.42264973)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $0.32264973$ in the expression.

$f'' (0.32264973) = 48 {(0.32264973)}^{2} - 96 \cdot 0.32264973 + 32$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $0.32264973$ to the power of $2$ .

$f'' (0.32264973) = 48 \cdot 0.10410284 - 96 \cdot 0.32264973 + 32$

Step 5.2.1.2

Multiply $48$ by $0.10410284$ .

$f'' (0.32264973) = 4.99693674 - 96 \cdot 0.32264973 + 32$

Step 5.2.1.3

Multiply $- 96$ by $0.32264973$ .

$f'' (0.32264973) = 4.99693674 - 30.97437415 + 32$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $30.97437415$ from $4.99693674$ .

$f'' (0.32264973) = - 25.97743741 + 32$

Step 5.2.2.2

Add $- 25.97743741$ and $32$ .

$f'' (0.32264973) = 6.02256258$

Step 5.2.3

The final answer is $6.02256258$ .

$6.02256258$

Step 5.3

At $0.32264973$ , the second derivative is $6.02256258$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0.42264973)$ .

Increasing on $(- \infty, 0.42264973)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(0.42264973, 1.57735026)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $1$ in the expression.

$f'' (1) = 48 {(1)}^{2} - 96 \cdot 1 + 32$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $1$ to the power of $2$ .

$f'' (1) = 48 \cdot 1 - 96 \cdot 1 + 32$

Step 6.2.1.2

Multiply $48$ by $1$ .

$f'' (1) = 48 - 96 \cdot 1 + 32$

Step 6.2.1.3

Multiply $- 96$ by $1$ .

$f'' (1) = 48 - 96 + 32$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $96$ from $48$ .

$f'' (1) = - 48 + 32$

Step 6.2.2.2

Add $- 48$ and $32$ .

$f'' (1) = - 16$

Step 6.2.3

The final answer is $- 16$ .

$- 16$

Step 6.3

At $1$ , the second derivative is $- 16$ . Since this is negative, the second derivative is decreasing on the interval $(0.42264973, 1.57735026)$

Decreasing on $(0.42264973, 1.57735026)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(1.57735026, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.67735026$ in the expression.

$f'' (1.67735026) = 48 {(1.67735026)}^{2} - 96 \cdot 1.67735026 + 32$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $1.67735026$ to the power of $2$ .

$f'' (1.67735026) = 48 \cdot 2.81350392 - 96 \cdot 1.67735026 + 32$

Step 7.2.1.2

Multiply $48$ by $2.81350392$ .

$f'' (1.67735026) = 135.04818842 - 96 \cdot 1.67735026 + 32$

Step 7.2.1.3

Multiply $- 96$ by $1.67735026$ .

$f'' (1.67735026) = 135.04818842 - 161.02562584 + 32$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $161.02562584$ from $135.04818842$ .

$f'' (1.67735026) = - 25.97743741 + 32$

Step 7.2.2.2

Add $- 25.97743741$ and $32$ .

$f'' (1.67735026) = 6.02256258$

Step 7.2.3

The final answer is $6.02256258$ .

$6.02256258$

Step 7.3

At $1.67735026$ , the second derivative is $6.02256258$ . Since this is positive, the second derivative is increasing on the interval $(1.57735026, \infty)$ .

Increasing on $(1.57735026, \infty)$ since $f'' (x) > 0$

Step 8

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 $(0.42264973, 1.77777777), (1.57735026, 1.77777777)$ .

$(0.42264973, 1.77777777), (1.57735026, 1.77777777)$

Step 9