Calculus Examples

$y = x^{4} - 8 x^{3} + 16 x^{2}$

Step 1

Write $y = x^{4} - 8 x^{3} + 16 x^{2}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate.

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Step 2.1.1.1

By the Sum Rule, the derivative of $x^{4} - 8 x^{3} + 16 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x^{2}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x^{2}]$

Step 2.1.1.2

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

$4 x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x^{2}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 2.1.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$4 x^{3} - 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [16 x^{2}]$

Step 2.1.2.2

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

$4 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} [16 x^{2}]$

Step 2.1.2.3

Multiply $3$ by $- 8$ .

$4 x^{3} - 24 x^{2} + \frac{d}{d x} [16 x^{2}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [16 x^{2}]$ .

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Step 2.1.3.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{2}$ with respect to $x$ is $16 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 24 x^{2} + 16 \frac{d}{d x} [x^{2}]$

Step 2.1.3.2

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

$4 x^{3} - 24 x^{2} + 16 (2 x)$

Step 2.1.3.3

Multiply $2$ by $16$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 32 x$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $4 x^{3} - 24 x^{2} + 32 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [32 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [32 x]$

Step 2.2.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 2.2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [32 x]$

Step 2.2.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [32 x]$

Step 2.2.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [32 x]$

Step 2.2.3

Evaluate $\frac{d}{d x} [- 24 x^{2}]$ .

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Step 2.2.3.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x^{2}$ with respect to $x$ is $- 24 \frac{d}{d x} [x^{2}]$ .

$12 x^{2} - 24 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [32 x]$

Step 2.2.3.2

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

$12 x^{2} - 24 (2 x) + \frac{d}{d x} [32 x]$

Step 2.2.3.3

Multiply $2$ by $- 24$ .

$12 x^{2} - 48 x + \frac{d}{d x} [32 x]$

Step 2.2.4

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

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Step 2.2.4.1

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

$12 x^{2} - 48 x + 32 \frac{d}{d x} [x]$

Step 2.2.4.2

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

$12 x^{2} - 48 x + 32 \cdot 1$

Step 2.2.4.3

Multiply $32$ by $1$ .

$f'' (x) = 12 x^{2} - 48 x + 32$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 48 x + 32$ .

$12 x^{2} - 48 x + 32$

Step 3

Set the second derivative equal to $0$ then solve the equation $12 x^{2} - 48 x + 32 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$12 x^{2} - 48 x + 32 = 0$

Step 3.2

Factor $4$ out of $12 x^{2} - 48 x + 32$ .

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Step 3.2.1

Factor $4$ out of $12 x^{2}$ .

$4 (3 x^{2}) - 48 x + 32 = 0$

Step 3.2.2

Factor $4$ out of $- 48 x$ .

$4 (3 x^{2}) + 4 (- 12 x) + 32 = 0$

Step 3.2.3

Factor $4$ out of $32$ .

$4 (3 x^{2}) + 4 (- 12 x) + 4 (8) = 0$

Step 3.2.4

Factor $4$ out of $4 (3 x^{2}) + 4 (- 12 x)$ .

$4 (3 x^{2} - 12 x) + 4 (8) = 0$

Step 3.2.5

Factor $4$ out of $4 (3 x^{2} - 12 x) + 4 (8)$ .

$4 (3 x^{2} - 12 x + 8) = 0$

Step 3.3

Divide each term in $4 (3 x^{2} - 12 x + 8) = 0$ by $4$ and simplify.

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Step 3.3.1

Divide each term in $4 (3 x^{2} - 12 x + 8) = 0$ by $4$ .

$\frac{4 (3 x^{2} - 12 x + 8)}{4} = \frac{0}{4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{4 (3 x^{2} - 12 x + 8)}{4} = \frac{0}{4}$

Step 3.3.2.1.2

Divide $3 x^{2} - 12 x + 8$ by $1$ .

$3 x^{2} - 12 x + 8 = \frac{0}{4}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $4$ .

$3 x^{2} - 12 x + 8 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 3$ , $b = - 12$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (3 \cdot 8)}}{2 \cdot 3}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot 8}}{2 \cdot 3}$

Step 3.6.1.2

Multiply $- 4 \cdot 3 \cdot 8$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 8}}{2 \cdot 3}$

Step 3.6.1.2.2

Multiply $- 12$ by $8$ .

$x = \frac{12 \pm \sqrt{144 - 96}}{2 \cdot 3}$

Step 3.6.1.3

Subtract $96$ from $144$ .

$x = \frac{12 \pm \sqrt{48}}{2 \cdot 3}$

Step 3.6.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 3.6.1.4.1

Factor $16$ out of $48$ .

$x = \frac{12 \pm \sqrt{16 (3)}}{2 \cdot 3}$

Step 3.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 3}$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{3}}{2 \cdot 3}$

Step 3.6.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm 4 \sqrt{3}}{6}$

Step 3.6.3

Simplify $\frac{12 \pm 4 \sqrt{3}}{6}$ .

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

Step 3.7

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

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot 8}}{2 \cdot 3}$

Step 3.7.1.2

Multiply $- 4 \cdot 3 \cdot 8$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 8}}{2 \cdot 3}$

Step 3.7.1.2.2

Multiply $- 12$ by $8$ .

$x = \frac{12 \pm \sqrt{144 - 96}}{2 \cdot 3}$

Step 3.7.1.3

Subtract $96$ from $144$ .

$x = \frac{12 \pm \sqrt{48}}{2 \cdot 3}$

Step 3.7.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 3.7.1.4.1

Factor $16$ out of $48$ .

$x = \frac{12 \pm \sqrt{16 (3)}}{2 \cdot 3}$

Step 3.7.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 3}$

Step 3.7.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{3}}{2 \cdot 3}$

Step 3.7.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm 4 \sqrt{3}}{6}$

Step 3.7.3

Simplify $\frac{12 \pm 4 \sqrt{3}}{6}$ .

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

Step 3.7.4

Change the $\pm$ to $+$ .

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

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot 8}}{2 \cdot 3}$

Step 3.8.1.2

Multiply $- 4 \cdot 3 \cdot 8$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 8}}{2 \cdot 3}$

Step 3.8.1.2.2

Multiply $- 12$ by $8$ .

$x = \frac{12 \pm \sqrt{144 - 96}}{2 \cdot 3}$

Step 3.8.1.3

Subtract $96$ from $144$ .

$x = \frac{12 \pm \sqrt{48}}{2 \cdot 3}$

Step 3.8.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 3.8.1.4.1

Factor $16$ out of $48$ .

$x = \frac{12 \pm \sqrt{16 (3)}}{2 \cdot 3}$

Step 3.8.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 3}$

Step 3.8.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{3}}{2 \cdot 3}$

Step 3.8.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm 4 \sqrt{3}}{6}$

Step 3.8.3

Simplify $\frac{12 \pm 4 \sqrt{3}}{6}$ .

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

Step 3.8.4

Change the $\pm$ to $-$ .

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

Step 3.9

The final answer is the combination of both solutions.

$x = \frac{6 + 2 \sqrt{3}}{3}, \frac{6 - 2 \sqrt{3}}{3}$

Step 4

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

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Step 4.1

Substitute $3.15470053$ in $f (x) = x^{4} - 8 x^{3} + 16 x^{2}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $3.15470053$ in the expression.

$f (3.15470053) = {(3.15470053)}^{4} - 8 {(3.15470053)}^{3} + 16 {(3.15470053)}^{2}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $3.15470053$ to the power of $4$ .

$f (3.15470053) = 99.04500074 - 8 {(3.15470053)}^{3} + 16 {(3.15470053)}^{2}$

Step 4.1.2.1.2

Raise $3.15470053$ to the power of $3$ .

$f (3.15470053) = 99.04500074 - 8 \cdot 31.39600717 + 16 {(3.15470053)}^{2}$

Step 4.1.2.1.3

Multiply $- 8$ by $31.39600717$ .

$f (3.15470053) = 99.04500074 - 251.16805742 + 16 {(3.15470053)}^{2}$

Step 4.1.2.1.4

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

$f (3.15470053) = 99.04500074 - 251.16805742 + 16 \cdot 9.95213548$

Step 4.1.2.1.5

Multiply $16$ by $9.95213548$ .

$f (3.15470053) = 99.04500074 - 251.16805742 + 159.23416778$

Step 4.1.2.2

Simplify by adding and subtracting.

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Step 4.1.2.2.1

Subtract $251.16805742$ from $99.04500074$ .

$f (3.15470053) = - 152.12305667 + 159.23416778$

Step 4.1.2.2.2

Add $- 152.12305667$ and $159.23416778$ .

$f (3.15470053) = 7.11111111$

Step 4.1.2.3

The final answer is $7.11111111$ .

$7.11111111$

Step 4.2

The point found by substituting $3.15470053$ in $f (x) = x^{4} - 8 x^{3} + 16 x^{2}$ is $(3.15470053, 7.11111111)$ . This point can be an inflection point.

$(3.15470053, 7.11111111)$

Step 4.3

Substitute $0.84529946$ in $f (x) = x^{4} - 8 x^{3} + 16 x^{2}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $0.84529946$ in the expression.

$f (0.84529946) = {(0.84529946)}^{4} - 8 {(0.84529946)}^{3} + 16 {(0.84529946)}^{2}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $0.84529946$ to the power of $4$ .

$f (0.84529946) = 0.5105548 - 8 {(0.84529946)}^{3} + 16 {(0.84529946)}^{2}$

Step 4.3.2.1.2

Raise $0.84529946$ to the power of $3$ .

$f (0.84529946) = 0.5105548 - 8 \cdot 0.60399282 + 16 {(0.84529946)}^{2}$

Step 4.3.2.1.3

Multiply $- 8$ by $0.60399282$ .

$f (0.84529946) = 0.5105548 - 4.83194257 + 16 {(0.84529946)}^{2}$

Step 4.3.2.1.4

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

$f (0.84529946) = 0.5105548 - 4.83194257 + 16 \cdot 0.71453117$

Step 4.3.2.1.5

Multiply $16$ by $0.71453117$ .

$f (0.84529946) = 0.5105548 - 4.83194257 + 11.43249887$

Step 4.3.2.2

Simplify by adding and subtracting.

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Step 4.3.2.2.1

Subtract $4.83194257$ from $0.5105548$ .

$f (0.84529946) = - 4.32138776 + 11.43249887$

Step 4.3.2.2.2

Add $- 4.32138776$ and $11.43249887$ .

$f (0.84529946) = 7.\overline{1}$

Step 4.3.2.3

The final answer is $7.\overline{1}$ .

$7.\overline{1}$

Step 4.4

The point found by substituting $0.84529946$ in $f (x) = x^{4} - 8 x^{3} + 16 x^{2}$ is $(0.84529946, 7.\overline{1})$ . This point can be an inflection point.

$(0.84529946, 7.\overline{1})$

Step 4.5

Determine the points that could be inflection points.

$(3.15470053, 7.11111111), (0.84529946, 7.\overline{1})$

Step 5

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

$(- \infty, 0.84529946) \cup (0.84529946, 3.15470053) \cup (3.15470053, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0.84529946)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.74529946$ in the expression.

$f'' (0.74529946) = 12 {(0.74529946)}^{2} - 48 \cdot 0.74529946 + 32$

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 $0.74529946$ to the power of $2$ .

$f'' (0.74529946) = 12 \cdot 0.55547128 - 48 \cdot 0.74529946 + 32$

Step 6.2.1.2

Multiply $12$ by $0.55547128$ .

$f'' (0.74529946) = 6.66565544 - 48 \cdot 0.74529946 + 32$

Step 6.2.1.3

Multiply $- 48$ by $0.74529946$ .

$f'' (0.74529946) = 6.66565544 - 35.77437415 + 32$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $35.77437415$ from $6.66565544$ .

$f'' (0.74529946) = - 29.1087187 + 32$

Step 6.2.2.2

Add $- 29.1087187$ and $32$ .

$f'' (0.74529946) = 2.89128129$

Step 6.2.3

The final answer is $2.89128129$ .

$2.89128129$

Step 6.3

At $0.74529946$ , the second derivative is $2.89128129$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0.84529946)$ .

Increasing on $(- \infty, 0.84529946)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(0.84529946, 3.15470053)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

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

$f'' (2) = 12 {(2)}^{2} - 48 \cdot 2 + 32$

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 $2$ to the power of $2$ .

$f'' (2) = 12 \cdot 4 - 48 \cdot 2 + 32$

Step 7.2.1.2

Multiply $12$ by $4$ .

$f'' (2) = 48 - 48 \cdot 2 + 32$

Step 7.2.1.3

Multiply $- 48$ by $2$ .

$f'' (2) = 48 - 96 + 32$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $96$ from $48$ .

$f'' (2) = - 48 + 32$

Step 7.2.2.2

Add $- 48$ and $32$ .

$f'' (2) = - 16$

Step 7.2.3

The final answer is $- 16$ .

$- 16$

Step 7.3

At $2$ , the second derivative is $- 16$ . Since this is negative, the second derivative is decreasing on the interval $(0.84529946, 3.15470053)$

Decreasing on $(0.84529946, 3.15470053)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(3.15470053, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $3.25470053$ in the expression.

$f'' (3.25470053) = 12 {(3.25470053)}^{2} - 48 \cdot 3.25470053 + 32$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $3.25470053$ to the power of $2$ .

$f'' (3.25470053) = 12 \cdot 10.59307559 - 48 \cdot 3.25470053 + 32$

Step 8.2.1.2

Multiply $12$ by $10.59307559$ .

$f'' (3.25470053) = 127.11690713 - 48 \cdot 3.25470053 + 32$

Step 8.2.1.3

Multiply $- 48$ by $3.25470053$ .

$f'' (3.25470053) = 127.11690713 - 156.22562584 + 32$

Step 8.2.2

Simplify by adding and subtracting.

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Step 8.2.2.1

Subtract $156.22562584$ from $127.11690713$ .

$f'' (3.25470053) = - 29.1087187 + 32$

Step 8.2.2.2

Add $- 29.1087187$ and $32$ .

$f'' (3.25470053) = 2.89128129$

Step 8.2.3

The final answer is $2.89128129$ .

$2.89128129$

Step 8.3

At $3.25470053$ , the second derivative is $2.89128129$ . Since this is positive, the second derivative is increasing on the interval $(3.15470053, \infty)$ .

Increasing on $(3.15470053, \infty)$ 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 $(0.84529946, 7.\overline{1}), (3.15470053, 7.11111111)$ .

$(0.84529946, 7.\overline{1}), (3.15470053, 7.11111111)$

Step 10