Calculus Examples

$f (x) = - 3 x^{4} + 28 x^{3} - 60 x^{2}$

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

By the Sum Rule, the derivative of $- 3 x^{4} + 28 x^{3} - 60 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [- 60 x^{2}]$ .

$\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [- 60 x^{2}]$

Step 1.1.2

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

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Step 1.1.2.1

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

$- 3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [- 60 x^{2}]$

Step 1.1.2.2

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

$- 3 (4 x^{3}) + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [- 60 x^{2}]$

Step 1.1.2.3

Multiply $4$ by $- 3$ .

$- 12 x^{3} + \frac{d}{d x} [28 x^{3}] + \frac{d}{d x} [- 60 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [28 x^{3}]$ .

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Step 1.1.3.1

Since $28$ is constant with respect to $x$ , the derivative of $28 x^{3}$ with respect to $x$ is $28 \frac{d}{d x} [x^{3}]$ .

$- 12 x^{3} + 28 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 60 x^{2}]$

Step 1.1.3.2

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

$- 12 x^{3} + 28 (3 x^{2}) + \frac{d}{d x} [- 60 x^{2}]$

Step 1.1.3.3

Multiply $3$ by $28$ .

$- 12 x^{3} + 84 x^{2} + \frac{d}{d x} [- 60 x^{2}]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 60 x^{2}]$ .

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Step 1.1.4.1

Since $- 60$ is constant with respect to $x$ , the derivative of $- 60 x^{2}$ with respect to $x$ is $- 60 \frac{d}{d x} [x^{2}]$ .

$- 12 x^{3} + 84 x^{2} - 60 \frac{d}{d x} [x^{2}]$

Step 1.1.4.2

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

$- 12 x^{3} + 84 x^{2} - 60 (2 x)$

Step 1.1.4.3

Multiply $2$ by $- 60$ .

$f' (x) = - 12 x^{3} + 84 x^{2} - 120 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $- 12 x^{3} + 84 x^{2} - 120 x$ with respect to $x$ is $\frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [84 x^{2}] + \frac{d}{d x} [- 120 x]$ .

$\frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [84 x^{2}] + \frac{d}{d x} [- 120 x]$

Step 1.2.2

Evaluate $\frac{d}{d x} [- 12 x^{3}]$ .

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Step 1.2.2.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{3}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{3}]$ .

$- 12 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [84 x^{2}] + \frac{d}{d x} [- 120 x]$

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 $n = 3$ .

$- 12 (3 x^{2}) + \frac{d}{d x} [84 x^{2}] + \frac{d}{d x} [- 120 x]$

Step 1.2.2.3

Multiply $3$ by $- 12$ .

$- 36 x^{2} + \frac{d}{d x} [84 x^{2}] + \frac{d}{d x} [- 120 x]$

Step 1.2.3

Evaluate $\frac{d}{d x} [84 x^{2}]$ .

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Step 1.2.3.1

Since $84$ is constant with respect to $x$ , the derivative of $84 x^{2}$ with respect to $x$ is $84 \frac{d}{d x} [x^{2}]$ .

$- 36 x^{2} + 84 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 120 x]$

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 $n = 2$ .

$- 36 x^{2} + 84 (2 x) + \frac{d}{d x} [- 120 x]$

Step 1.2.3.3

Multiply $2$ by $84$ .

$- 36 x^{2} + 168 x + \frac{d}{d x} [- 120 x]$

Step 1.2.4

Evaluate $\frac{d}{d x} [- 120 x]$ .

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Step 1.2.4.1

Since $- 120$ is constant with respect to $x$ , the derivative of $- 120 x$ with respect to $x$ is $- 120 \frac{d}{d x} [x]$ .

$- 36 x^{2} + 168 x - 120 \frac{d}{d x} [x]$

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 $n = 1$ .

$- 36 x^{2} + 168 x - 120 \cdot 1$

Step 1.2.4.3

Multiply $- 120$ by $1$ .

$f'' (x) = - 36 x^{2} + 168 x - 120$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- 36 x^{2} + 168 x - 120$ .

$- 36 x^{2} + 168 x - 120$

Step 2

Set the second derivative equal to $0$ then solve the equation $- 36 x^{2} + 168 x - 120 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- 36 x^{2} + 168 x - 120 = 0$

Step 2.2

Factor $- 12$ out of $- 36 x^{2} + 168 x - 120$ .

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Step 2.2.1

Factor $- 12$ out of $- 36 x^{2}$ .

$- 12 (3 x^{2}) + 168 x - 120 = 0$

Step 2.2.2

Factor $- 12$ out of $168 x$ .

$- 12 (3 x^{2}) - 12 (- 14 x) - 120 = 0$

Step 2.2.3

Factor $- 12$ out of $- 120$ .

$- 12 (3 x^{2}) - 12 (- 14 x) - 12 \cdot 10 = 0$

Step 2.2.4

Factor $- 12$ out of $- 12 (3 x^{2}) - 12 (- 14 x)$ .

$- 12 (3 x^{2} - 14 x) - 12 \cdot 10 = 0$

Step 2.2.5

Factor $- 12$ out of $- 12 (3 x^{2} - 14 x) - 12 (10)$ .

$- 12 (3 x^{2} - 14 x + 10) = 0$

Step 2.3

Divide each term in $- 12 (3 x^{2} - 14 x + 10) = 0$ by $- 12$ and simplify.

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Step 2.3.1

Divide each term in $- 12 (3 x^{2} - 14 x + 10) = 0$ by $- 12$ .

$\frac{- 12 (3 x^{2} - 14 x + 10)}{- 12} = \frac{0}{- 12}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 12 (3 x^{2} - 14 x + 10)}{- 12} = \frac{0}{- 12}$

Step 2.3.2.1.2

Divide $3 x^{2} - 14 x + 10$ by $1$ .

$3 x^{2} - 14 x + 10 = \frac{0}{- 12}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $- 12$ .

$3 x^{2} - 14 x + 10 = 0$

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 $a = 3$ , $b = - 14$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (3 \cdot 10)}}{2 \cdot 3}$

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 $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}$

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot 10$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 10}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply $- 12$ by $10$ .

$x = \frac{14 \pm \sqrt{196 - 120}}{2 \cdot 3}$

Step 2.6.1.3

Subtract $120$ from $196$ .

$x = \frac{14 \pm \sqrt{76}}{2 \cdot 3}$

Step 2.6.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 2.6.1.4.1

Factor $4$ out of $76$ .

$x = \frac{14 \pm \sqrt{4 (19)}}{2 \cdot 3}$

Step 2.6.1.4.2

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{19}}{2 \cdot 3}$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{19}}{6}$

Step 2.6.3

Simplify $\frac{14 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{7 \pm \sqrt{19}}{3}$

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 $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}$

Step 2.7.1.2

Multiply $- 4 \cdot 3 \cdot 10$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 10}}{2 \cdot 3}$

Step 2.7.1.2.2

Multiply $- 12$ by $10$ .

$x = \frac{14 \pm \sqrt{196 - 120}}{2 \cdot 3}$

Step 2.7.1.3

Subtract $120$ from $196$ .

$x = \frac{14 \pm \sqrt{76}}{2 \cdot 3}$

Step 2.7.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 2.7.1.4.1

Factor $4$ out of $76$ .

$x = \frac{14 \pm \sqrt{4 (19)}}{2 \cdot 3}$

Step 2.7.1.4.2

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{19}}{2 \cdot 3}$

Step 2.7.2

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{19}}{6}$

Step 2.7.3

Simplify $\frac{14 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{7 \pm \sqrt{19}}{3}$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{19}}{3}$

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 $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}$

Step 2.8.1.2

Multiply $- 4 \cdot 3 \cdot 10$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 10}}{2 \cdot 3}$

Step 2.8.1.2.2

Multiply $- 12$ by $10$ .

$x = \frac{14 \pm \sqrt{196 - 120}}{2 \cdot 3}$

Step 2.8.1.3

Subtract $120$ from $196$ .

$x = \frac{14 \pm \sqrt{76}}{2 \cdot 3}$

Step 2.8.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 2.8.1.4.1

Factor $4$ out of $76$ .

$x = \frac{14 \pm \sqrt{4 (19)}}{2 \cdot 3}$

Step 2.8.1.4.2

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 3}$

Step 2.8.1.5

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{19}}{2 \cdot 3}$

Step 2.8.2

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{19}}{6}$

Step 2.8.3

Simplify $\frac{14 \pm 2 \sqrt{19}}{6}$ .

$x = \frac{7 \pm \sqrt{19}}{3}$

Step 2.8.4

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{19}}{3}$

Step 2.9

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{19}}{3}, \frac{7 - \sqrt{19}}{3}$

Step 3

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

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Step 3.1

Substitute $3.78629964$ in $f (x) = - 3 x^{4} + 28 x^{3} - 60 x^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $3.78629964$ in the expression.

$f (3.78629964) = - 3 {(3.78629964)}^{4} + 28 {(3.78629964)}^{3} - 60 {(3.78629964)}^{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Raise $3.78629964$ to the power of $4$ .

$f (3.78629964) = - 3 \cdot 205.52276035 + 28 {(3.78629964)}^{3} - 60 {(3.78629964)}^{2}$

Step 3.1.2.1.2

Multiply $- 3$ by $205.52276035$ .

$f (3.78629964) = - 616.56828105 + 28 {(3.78629964)}^{3} - 60 {(3.78629964)}^{2}$

Step 3.1.2.1.3

Raise $3.78629964$ to the power of $3$ .

$f (3.78629964) = - 616.56828105 + 28 \cdot 54.28063794 - 60 {(3.78629964)}^{2}$

Step 3.1.2.1.4

Multiply $28$ by $54.28063794$ .

$f (3.78629964) = - 616.56828105 + 1519.85786257 - 60 {(3.78629964)}^{2}$

Step 3.1.2.1.5

Raise $3.78629964$ to the power of $2$ .

$f (3.78629964) = - 616.56828105 + 1519.85786257 - 60 \cdot 14.33606502$

Step 3.1.2.1.6

Multiply $- 60$ by $14.33606502$ .

$f (3.78629964) = - 616.56828105 + 1519.85786257 - 860.16390139$

Step 3.1.2.2

Simplify by adding and subtracting.

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Step 3.1.2.2.1

Add $- 616.56828105$ and $1519.85786257$ .

$f (3.78629964) = 903.28958152 - 860.16390139$

Step 3.1.2.2.2

Subtract $860.16390139$ from $903.28958152$ .

$f (3.78629964) = 43.12568012$

Step 3.1.2.3

The final answer is $43.12568012$ .

$43.12568012$

Step 3.2

The point found by substituting $3.78629964$ in $f (x) = - 3 x^{4} + 28 x^{3} - 60 x^{2}$ is $(3.78629964, 43.12568012)$ . This point can be an inflection point.

$(3.78629964, 43.12568012)$

Step 3.3

Substitute $0.88036701$ in $f (x) = - 3 x^{4} + 28 x^{3} - 60 x^{2}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $0.88036701$ in the expression.

$f (0.88036701) = - 3 {(0.88036701)}^{4} + 28 {(0.88036701)}^{3} - 60 {(0.88036701)}^{2}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Raise $0.88036701$ to the power of $4$ .

$f (0.88036701) = - 3 \cdot 0.60069643 + 28 {(0.88036701)}^{3} - 60 {(0.88036701)}^{2}$

Step 3.3.2.1.2

Multiply $- 3$ by $0.60069643$ .

$f (0.88036701) = - 1.80208931 + 28 {(0.88036701)}^{3} - 60 {(0.88036701)}^{2}$

Step 3.3.2.1.3

Raise $0.88036701$ to the power of $3$ .

$f (0.88036701) = - 1.80208931 + 28 \cdot 0.68232501 - 60 {(0.88036701)}^{2}$

Step 3.3.2.1.4

Multiply $28$ by $0.68232501$ .

$f (0.88036701) = - 1.80208931 + 19.10510038 - 60 {(0.88036701)}^{2}$

Step 3.3.2.1.5

Raise $0.88036701$ to the power of $2$ .

$f (0.88036701) = - 1.80208931 + 19.10510038 - 60 \cdot 0.77504608$

Step 3.3.2.1.6

Multiply $- 60$ by $0.77504608$ .

$f (0.88036701) = - 1.80208931 + 19.10510038 - 46.50276526$

Step 3.3.2.2

Simplify by adding and subtracting.

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Step 3.3.2.2.1

Add $- 1.80208931$ and $19.10510038$ .

$f (0.88036701) = 17.30301107 - 46.50276526$

Step 3.3.2.2.2

Subtract $46.50276526$ from $17.30301107$ .

$f (0.88036701) = - 29.19975419$

Step 3.3.2.3

The final answer is $- 29.19975419$ .

$- 29.19975419$

Step 3.4

The point found by substituting $0.88036701$ in $f (x) = - 3 x^{4} + 28 x^{3} - 60 x^{2}$ is $(0.88036701, - 29.19975419)$ . This point can be an inflection point.

$(0.88036701, - 29.19975419)$

Step 3.5

Determine the points that could be inflection points.

$(3.78629964, 43.12568012), (0.88036701, - 29.19975419)$

Step 4

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

$(- \infty, 0.88036701) \cup (0.88036701, 3.78629964) \cup (3.78629964, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0.88036701)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $0.78036701$ in the expression.

$f'' (0.78036701) = - 36 {(0.78036701)}^{2} + 168 (0.78036701) - 120$

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 $0.78036701$ to the power of $2$ .

$f'' (0.78036701) = - 36 \cdot 0.60897268 + 168 (0.78036701) - 120$

Step 5.2.1.2

Multiply $- 36$ by $0.60897268$ .

$f'' (0.78036701) = - 21.92301662 + 168 (0.78036701) - 120$

Step 5.2.1.3

Multiply $168$ by $0.78036701$ .

$f'' (0.78036701) = - 21.92301662 + 131.10165916 - 120$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 21.92301662$ and $131.10165916$ .

$f'' (0.78036701) = 109.17864253 - 120$

Step 5.2.2.2

Subtract $120$ from $109.17864253$ .

$f'' (0.78036701) = - 10.82135746$

Step 5.2.3

The final answer is $- 10.82135746$ .

$- 10.82135746$

Step 5.3

At $0.78036701$ , the second derivative is $- 10.82135746$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, 0.88036701)$

Decreasing on $(- \infty, 0.88036701)$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(0.88036701, 3.78629964)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $2.33333333$ in the expression.

$f'' (2.33333333) = - 36 {(2.33333333)}^{2} + 168 (2.33333333) - 120$

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 $2.33333333$ to the power of $2$ .

$f'' (2.33333333) = - 36 \cdot 5.\overline{4} + 168 (2.33333333) - 120$

Step 6.2.1.2

Multiply $- 36$ by $5.\overline{4}$ .

$f'' (2.33333333) = - 196 + 168 (2.33333333) - 120$

Step 6.2.1.3

Multiply $168$ by $2.33333333$ .

$f'' (2.33333333) = - 196 + 392 - 120$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 196$ and $392$ .

$f'' (2.33333333) = 196 - 120$

Step 6.2.2.2

Subtract $120$ from $196$ .

$f'' (2.33333333) = 76$

Step 6.2.3

The final answer is $76$ .

$76$

Step 6.3

At $2.33333333$ , the second derivative is $76$ . Since this is positive, the second derivative is increasing on the interval $(0.88036701, 3.78629964)$ .

Increasing on $(0.88036701, 3.78629964)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(3.78629964, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3.88629964$ in the expression.

$f'' (3.88629964) = - 36 {(3.88629964)}^{2} + 168 (3.88629964) - 120$

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 $3.88629964$ to the power of $2$ .

$f'' (3.88629964) = - 36 \cdot 15.10332495 + 168 (3.88629964) - 120$

Step 7.2.1.2

Multiply $- 36$ by $15.10332495$ .

$f'' (3.88629964) = - 543.7196983 + 168 (3.88629964) - 120$

Step 7.2.1.3

Multiply $168$ by $3.88629964$ .

$f'' (3.88629964) = - 543.7196983 + 652.89834083 - 120$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $- 543.7196983$ and $652.89834083$ .

$f'' (3.88629964) = 109.17864253 - 120$

Step 7.2.2.2

Subtract $120$ from $109.17864253$ .

$f'' (3.88629964) = - 10.82135746$

Step 7.2.3

The final answer is $- 10.82135746$ .

$- 10.82135746$

Step 7.3

At $3.88629964$ , the second derivative is $- 10.82135746$ . Since this is negative, the second derivative is decreasing on the interval $(3.78629964, \infty)$

Decreasing on $(3.78629964, \infty)$ 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 $(0.88036701, - 29.19975419), (3.78629964, 43.12568012)$ .

$(0.88036701, - 29.19975419), (3.78629964, 43.12568012)$

Step 9