Calculus Examples

$f (x) = \frac{1}{6} x^{6} + x^{5} - 5 x^{4}$

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 $\frac{1}{6} x^{6} + x^{5} - 5 x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{6} x^{6}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{4}]$ .

$\frac{d}{d x} [\frac{1}{6} x^{6}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{4}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{6} x^{6}]$ .

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Step 1.1.2.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6} x^{6}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{6}]$ .

$\frac{1}{6} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{4}]$

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 $n = 6$ .

$\frac{1}{6} (6 x^{5}) + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{4}]$

Step 1.1.2.3

Combine $6$ and $\frac{1}{6}$ .

$\frac{6}{6} x^{5} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{4}]$

Step 1.1.2.4

Combine $\frac{6}{6}$ and $x^{5}$ .

$\frac{6 x^{5}}{6} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{4}]$

Step 1.1.2.5

Cancel the common factor of $6$ .

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Step 1.1.2.5.1

Cancel the common factor.

$\frac{6 x^{5}}{6} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{4}]$

Step 1.1.2.5.2

Divide $x^{5}$ by $1$ .

$x^{5} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{4}]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$x^{5} + 5 x^{4} + \frac{d}{d x} [- 5 x^{4}]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 5 x^{4}]$ .

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Step 1.1.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{4}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{4}]$ .

$x^{5} + 5 x^{4} - 5 \frac{d}{d x} [x^{4}]$

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 $n = 4$ .

$x^{5} + 5 x^{4} - 5 (4 x^{3})$

Step 1.1.4.3

Multiply $4$ by $- 5$ .

$f' (x) = x^{5} + 5 x^{4} - 20 x^{3}$

Step 1.2

Find the second derivative.

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Step 1.2.1

Differentiate.

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Step 1.2.1.1

By the Sum Rule, the derivative of $x^{5} + 5 x^{4} - 20 x^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 20 x^{3}]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 20 x^{3}]$

Step 1.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 20 x^{3}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 1.2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 x^{4} + 5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 20 x^{3}]$

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 $n = 4$ .

$5 x^{4} + 5 (4 x^{3}) + \frac{d}{d x} [- 20 x^{3}]$

Step 1.2.2.3

Multiply $4$ by $5$ .

$5 x^{4} + 20 x^{3} + \frac{d}{d x} [- 20 x^{3}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 20 x^{3}]$ .

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Step 1.2.3.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 x^{3}$ with respect to $x$ is $- 20 \frac{d}{d x} [x^{3}]$ .

$5 x^{4} + 20 x^{3} - 20 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

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

Step 1.2.3.3

Multiply $3$ by $- 20$ .

$f'' (x) = 5 x^{4} + 20 x^{3} - 60 x^{2}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $5 x^{4} + 20 x^{3} - 60 x^{2}$ .

$5 x^{4} + 20 x^{3} - 60 x^{2}$

Step 2

Set the second derivative equal to $0$ then solve the equation $5 x^{4} + 20 x^{3} - 60 x^{2} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$5 x^{4} + 20 x^{3} - 60 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $5 x^{2}$ out of $5 x^{4} + 20 x^{3} - 60 x^{2}$ .

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Step 2.2.1.1

Factor $5 x^{2}$ out of $5 x^{4}$ .

$5 x^{2} (x^{2}) + 20 x^{3} - 60 x^{2} = 0$

Step 2.2.1.2

Factor $5 x^{2}$ out of $20 x^{3}$ .

$5 x^{2} (x^{2}) + 5 x^{2} (4 x) - 60 x^{2} = 0$

Step 2.2.1.3

Factor $5 x^{2}$ out of $- 60 x^{2}$ .

$5 x^{2} (x^{2}) + 5 x^{2} (4 x) + 5 x^{2} (- 12) = 0$

Step 2.2.1.4

Factor $5 x^{2}$ out of $5 x^{2} (x^{2}) + 5 x^{2} (4 x)$ .

$5 x^{2} (x^{2} + 4 x) + 5 x^{2} (- 12) = 0$

Step 2.2.1.5

Factor $5 x^{2}$ out of $5 x^{2} (x^{2} + 4 x) + 5 x^{2} (- 12)$ .

$5 x^{2} (x^{2} + 4 x - 12) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} + 4 x - 12$ using the AC method.

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Step 2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $4$ .

$- 2, 6$

Step 2.2.2.1.2

Write the factored form using these integers.

$5 x^{2} ((x - 2) (x + 6)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$5 x^{2} (x - 2) (x + 6) = 0$

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 $0$ .

$x^{2} = 0$

$x - 2 = 0$

$x + 6 = 0$

Step 2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

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

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Step 2.5.1

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

$x - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.6

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 2.6.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 2.7

The final solution is all the values that make $5 x^{2} (x - 2) (x + 6) = 0$ true.

$x = 0, 2, - 6$

Step 3

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

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Step 3.1

Substitute $0$ in $f (x) = \frac{1}{6} x^{6} + x^{5} - 5 x^{4}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = \frac{1}{6} \cdot {(0)}^{6} + {(0)}^{5} - 5 {(0)}^{4}$

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

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

$f (0) = \frac{1}{6} \cdot 0 + {(0)}^{5} - 5 {(0)}^{4}$

Step 3.1.2.1.2

Multiply $\frac{1}{6}$ by $0$ .

$f (0) = 0 + {(0)}^{5} - 5 {(0)}^{4}$

Step 3.1.2.1.3

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

$f (0) = 0 + 0 - 5 {(0)}^{4}$

Step 3.1.2.1.4

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

$f (0) = 0 + 0 - 5 \cdot 0$

Step 3.1.2.1.5

Multiply $- 5$ by $0$ .

$f (0) = 0 + 0 + 0$

Step 3.1.2.2

Simplify by adding numbers.

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Step 3.1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 3.1.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.1.2.3

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = \frac{1}{6} x^{6} + x^{5} - 5 x^{4}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $2$ in $f (x) = \frac{1}{6} x^{6} + x^{5} - 5 x^{4}$ to find the value of $y$ .

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Step 3.3.1

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

$f (2) = \frac{1}{6} \cdot {(2)}^{6} + {(2)}^{5} - 5 {(2)}^{4}$

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 $2$ to the power of $6$ .

$f (2) = \frac{1}{6} \cdot 64 + {(2)}^{5} - 5 {(2)}^{4}$

Step 3.3.2.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.1

Factor $2$ out of $6$ .

$f (2) = \frac{1}{2 (3)} \cdot 64 + {(2)}^{5} - 5 {(2)}^{4}$

Step 3.3.2.1.2.2

Factor $2$ out of $64$ .

$f (2) = \frac{1}{2 \cdot 3} \cdot (2 \cdot 32) + {(2)}^{5} - 5 {(2)}^{4}$

Step 3.3.2.1.2.3

Cancel the common factor.

$f (2) = \frac{1}{2 \cdot 3} \cdot (2 \cdot 32) + {(2)}^{5} - 5 {(2)}^{4}$

Step 3.3.2.1.2.4

Rewrite the expression.

$f (2) = \frac{1}{3} \cdot 32 + {(2)}^{5} - 5 {(2)}^{4}$

Step 3.3.2.1.3

Combine $\frac{1}{3}$ and $32$ .

$f (2) = \frac{32}{3} + {(2)}^{5} - 5 {(2)}^{4}$

Step 3.3.2.1.4

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

$f (2) = \frac{32}{3} + 32 - 5 {(2)}^{4}$

Step 3.3.2.1.5

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

$f (2) = \frac{32}{3} + 32 - 5 \cdot 16$

Step 3.3.2.1.6

Multiply $- 5$ by $16$ .

$f (2) = \frac{32}{3} + 32 - 80$

Step 3.3.2.2

Find the common denominator.

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Step 3.3.2.2.1

Write $32$ as a fraction with denominator $1$ .

$f (2) = \frac{32}{3} + \frac{32}{1} - 80$

Step 3.3.2.2.2

Multiply $\frac{32}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{32}{3} + \frac{32}{1} \cdot \frac{3}{3} - 80$

Step 3.3.2.2.3

Multiply $\frac{32}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{32}{3} + \frac{32 \cdot 3}{3} - 80$

Step 3.3.2.2.4

Write $- 80$ as a fraction with denominator $1$ .

$f (2) = \frac{32}{3} + \frac{32 \cdot 3}{3} + \frac{- 80}{1}$

Step 3.3.2.2.5

Multiply $\frac{- 80}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{32}{3} + \frac{32 \cdot 3}{3} + \frac{- 80}{1} \cdot \frac{3}{3}$

Step 3.3.2.2.6

Multiply $\frac{- 80}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{32}{3} + \frac{32 \cdot 3}{3} + \frac{- 80 \cdot 3}{3}$

Step 3.3.2.3

Combine the numerators over the common denominator.

$f (2) = \frac{32 + 32 \cdot 3 - 80 \cdot 3}{3}$

Step 3.3.2.4

Simplify each term.

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Step 3.3.2.4.1

Multiply $32$ by $3$ .

$f (2) = \frac{32 + 96 - 80 \cdot 3}{3}$

Step 3.3.2.4.2

Multiply $- 80$ by $3$ .

$f (2) = \frac{32 + 96 - 240}{3}$

Step 3.3.2.5

Simplify the expression.

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Step 3.3.2.5.1

Add $32$ and $96$ .

$f (2) = \frac{128 - 240}{3}$

Step 3.3.2.5.2

Subtract $240$ from $128$ .

$f (2) = \frac{- 112}{3}$

Step 3.3.2.5.3

Move the negative in front of the fraction.

$f (2) = - \frac{112}{3}$

Step 3.3.2.6

The final answer is $- \frac{112}{3}$ .

$- \frac{112}{3}$

Step 3.4

The point found by substituting $2$ in $f (x) = \frac{1}{6} x^{6} + x^{5} - 5 x^{4}$ is $(2, - \frac{112}{3})$ . This point can be an inflection point.

$(2, - \frac{112}{3})$

Step 3.5

Substitute $- 6$ in $f (x) = \frac{1}{6} x^{6} + x^{5} - 5 x^{4}$ to find the value of $y$ .

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Step 3.5.1

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = \frac{1}{6} \cdot {(- 6)}^{6} + {(- 6)}^{5} - 5 {(- 6)}^{4}$

Step 3.5.2

Simplify the result.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

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

$f (- 6) = \frac{1}{6} \cdot 46656 + {(- 6)}^{5} - 5 {(- 6)}^{4}$

Step 3.5.2.1.2

Cancel the common factor of $6$ .

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Step 3.5.2.1.2.1

Factor $6$ out of $46656$ .

$f (- 6) = \frac{1}{6} \cdot (6 (7776)) + {(- 6)}^{5} - 5 {(- 6)}^{4}$

Step 3.5.2.1.2.2

Cancel the common factor.

$f (- 6) = \frac{1}{6} \cdot (6 \cdot 7776) + {(- 6)}^{5} - 5 {(- 6)}^{4}$

Step 3.5.2.1.2.3

Rewrite the expression.

$f (- 6) = 7776 + {(- 6)}^{5} - 5 {(- 6)}^{4}$

Step 3.5.2.1.3

Raise $- 6$ to the power of $5$ .

$f (- 6) = 7776 - 7776 - 5 {(- 6)}^{4}$

Step 3.5.2.1.4

Raise $- 6$ to the power of $4$ .

$f (- 6) = 7776 - 7776 - 5 \cdot 1296$

Step 3.5.2.1.5

Multiply $- 5$ by $1296$ .

$f (- 6) = 7776 - 7776 - 6480$

Step 3.5.2.2

Simplify by subtracting numbers.

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Step 3.5.2.2.1

Subtract $7776$ from $7776$ .

$f (- 6) = 0 - 6480$

Step 3.5.2.2.2

Subtract $6480$ from $0$ .

$f (- 6) = - 6480$

Step 3.5.2.3

The final answer is $- 6480$ .

$- 6480$

Step 3.6

The point found by substituting $- 6$ in $f (x) = \frac{1}{6} x^{6} + x^{5} - 5 x^{4}$ is $(- 6, - 6480)$ . This point can be an inflection point.

$(- 6, - 6480)$

Step 3.7

Determine the points that could be inflection points.

$(0, 0), (2, - \frac{112}{3}), (- 6, - 6480)$

Step 4

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

$(- \infty, - 6) \cup (- 6, 0) \cup (0, 2) \cup (2, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 6)$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 6.1$ in the expression.

$f'' (- 6.1) = 5 {(- 6.1)}^{4} + 20 {(- 6.1)}^{3} - 60 {(- 6.1)}^{2}$

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 $- 6.1$ to the power of $4$ .

$f'' (- 6.1) = 5 \cdot 1384.5841 + 20 {(- 6.1)}^{3} - 60 {(- 6.1)}^{2}$

Step 5.2.1.2

Multiply $5$ by $1384.5841$ .

$f'' (- 6.1) = 6922.9205 + 20 {(- 6.1)}^{3} - 60 {(- 6.1)}^{2}$

Step 5.2.1.3

Raise $- 6.1$ to the power of $3$ .

$f'' (- 6.1) = 6922.9205 + 20 \cdot - 226.981 - 60 {(- 6.1)}^{2}$

Step 5.2.1.4

Multiply $20$ by $- 226.981$ .

$f'' (- 6.1) = 6922.9205 - 4539.62 - 60 {(- 6.1)}^{2}$

Step 5.2.1.5

Raise $- 6.1$ to the power of $2$ .

$f'' (- 6.1) = 6922.9205 - 4539.62 - 60 \cdot 37.21$

Step 5.2.1.6

Multiply $- 60$ by $37.21$ .

$f'' (- 6.1) = 6922.9205 - 4539.62 - 2232.6$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $4539.62$ from $6922.9205$ .

$f'' (- 6.1) = 2383.3005 - 2232.6$

Step 5.2.2.2

Subtract $2232.6$ from $2383.3005$ .

$f'' (- 6.1) = 150.7005$

Step 5.2.3

The final answer is $150.7005$ .

$150.7005$

Step 5.3

At $- 6.1$ , the second derivative is $150.7005$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 6)$ .

Increasing on $(- \infty, - 6)$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- 6, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

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

$f'' (- 3) = 5 {(- 3)}^{4} + 20 {(- 3)}^{3} - 60 {(- 3)}^{2}$

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

$f'' (- 3) = 5 \cdot 81 + 20 {(- 3)}^{3} - 60 {(- 3)}^{2}$

Step 6.2.1.2

Multiply $5$ by $81$ .

$f'' (- 3) = 405 + 20 {(- 3)}^{3} - 60 {(- 3)}^{2}$

Step 6.2.1.3

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

$f'' (- 3) = 405 + 20 \cdot - 27 - 60 {(- 3)}^{2}$

Step 6.2.1.4

Multiply $20$ by $- 27$ .

$f'' (- 3) = 405 - 540 - 60 {(- 3)}^{2}$

Step 6.2.1.5

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

$f'' (- 3) = 405 - 540 - 60 \cdot 9$

Step 6.2.1.6

Multiply $- 60$ by $9$ .

$f'' (- 3) = 405 - 540 - 540$

Step 6.2.2

Simplify by subtracting numbers.

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Step 6.2.2.1

Subtract $540$ from $405$ .

$f'' (- 3) = - 135 - 540$

Step 6.2.2.2

Subtract $540$ from $- 135$ .

$f'' (- 3) = - 675$

Step 6.2.3

The final answer is $- 675$ .

$- 675$

Step 6.3

At $- 3$ , the second derivative is $- 675$ . Since this is negative, the second derivative is decreasing on the interval $(- 6, 0)$

Decreasing on $(- 6, 0)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0, 2)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

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

$f'' (1) = 5 {(1)}^{4} + 20 {(1)}^{3} - 60 {(1)}^{2}$

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

One to any power is one.

$f'' (1) = 5 \cdot 1 + 20 {(1)}^{3} - 60 {(1)}^{2}$

Step 7.2.1.2

Multiply $5$ by $1$ .

$f'' (1) = 5 + 20 {(1)}^{3} - 60 {(1)}^{2}$

Step 7.2.1.3

One to any power is one.

$f'' (1) = 5 + 20 \cdot 1 - 60 {(1)}^{2}$

Step 7.2.1.4

Multiply $20$ by $1$ .

$f'' (1) = 5 + 20 - 60 {(1)}^{2}$

Step 7.2.1.5

One to any power is one.

$f'' (1) = 5 + 20 - 60 \cdot 1$

Step 7.2.1.6

Multiply $- 60$ by $1$ .

$f'' (1) = 5 + 20 - 60$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $5$ and $20$ .

$f'' (1) = 25 - 60$

Step 7.2.2.2

Subtract $60$ from $25$ .

$f'' (1) = - 35$

Step 7.2.3

The final answer is $- 35$ .

$- 35$

Step 7.3

At $1$ , the second derivative is $- 35$ . Since this is negative, the second derivative is decreasing on the interval $(0, 2)$

Decreasing on $(0, 2)$ since $f'' (x) < 0$

Step 8

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

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Step 8.1

Replace the variable $x$ with $2.1$ in the expression.

$f'' (2.1) = 5 {(2.1)}^{4} + 20 {(2.1)}^{3} - 60 {(2.1)}^{2}$

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 $2.1$ to the power of $4$ .

$f'' (2.1) = 5 \cdot 19.4481 + 20 {(2.1)}^{3} - 60 {(2.1)}^{2}$

Step 8.2.1.2

Multiply $5$ by $19.4481$ .

$f'' (2.1) = 97.2405 + 20 {(2.1)}^{3} - 60 {(2.1)}^{2}$

Step 8.2.1.3

Raise $2.1$ to the power of $3$ .

$f'' (2.1) = 97.2405 + 20 \cdot 9.261 - 60 {(2.1)}^{2}$

Step 8.2.1.4

Multiply $20$ by $9.261$ .

$f'' (2.1) = 97.2405 + 185.22 - 60 {(2.1)}^{2}$

Step 8.2.1.5

Raise $2.1$ to the power of $2$ .

$f'' (2.1) = 97.2405 + 185.22 - 60 \cdot 4.41$

Step 8.2.1.6

Multiply $- 60$ by $4.41$ .

$f'' (2.1) = 97.2405 + 185.22 - 264.6$

Step 8.2.2

Simplify by adding and subtracting.

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Step 8.2.2.1

Add $97.2405$ and $185.22$ .

$f'' (2.1) = 282.4605 - 264.6$

Step 8.2.2.2

Subtract $264.6$ from $282.4605$ .

$f'' (2.1) = 17.8605$

Step 8.2.3

The final answer is $17.8605$ .

$17.8605$

Step 8.3

At $2.1$ , the second derivative is $17.8605$ . Since this is positive, the second derivative is increasing on the interval $(2, \infty)$ .

Increasing on $(2, \infty)$ 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 $(- 6, - 6480), (2, - \frac{112}{3})$ .

$(- 6, - 6480), (2, - \frac{112}{3})$

Step 10