Calculus Examples

$f (x) = - 12 x^{5} + 135 x^{4} - 400 x^{3}$

Step 1

Find the first 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 $- 12 x^{5} + 135 x^{4} - 400 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- 12 x^{5}] + \frac{d}{d x} [135 x^{4}] + \frac{d}{d x} [- 400 x^{3}]$ .

$f' (x) = \frac{d}{d x} (- 12 x^{5}) + \frac{d}{d x} (135 x^{4}) + \frac{d}{d x} (- 400 x^{3})$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 12 x^{5}]$ .

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Step 1.1.2.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{5}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{5}]$ .

$f' (x) = - 12 \frac{d}{d x} x^{5} + \frac{d}{d x} (135 x^{4}) + \frac{d}{d x} (- 400 x^{3})$

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 $n = 5$ .

$f' (x) = - 12 (5 x^{4}) + \frac{d}{d x} (135 x^{4}) + \frac{d}{d x} (- 400 x^{3})$

Step 1.1.2.3

Multiply $5$ by $- 12$ .

$f' (x) = - 60 x^{4} + \frac{d}{d x} (135 x^{4}) + \frac{d}{d x} (- 400 x^{3})$

Step 1.1.3

Evaluate $\frac{d}{d x} [135 x^{4}]$ .

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Step 1.1.3.1

Since $135$ is constant with respect to $x$ , the derivative of $135 x^{4}$ with respect to $x$ is $135 \frac{d}{d x} [x^{4}]$ .

$f' (x) = - 60 x^{4} + 135 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 400 x^{3})$

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 $n = 4$ .

$f' (x) = - 60 x^{4} + 135 (4 x^{3}) + \frac{d}{d x} (- 400 x^{3})$

Step 1.1.3.3

Multiply $4$ by $135$ .

$f' (x) = - 60 x^{4} + 540 x^{3} + \frac{d}{d x} (- 400 x^{3})$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 400 x^{3}]$ .

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Step 1.1.4.1

Since $- 400$ is constant with respect to $x$ , the derivative of $- 400 x^{3}$ with respect to $x$ is $- 400 \frac{d}{d x} [x^{3}]$ .

$f' (x) = - 60 x^{4} + 540 x^{3} - 400 \frac{d}{d x} x^{3}$

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 $n = 3$ .

$f' (x) = - 60 x^{4} + 540 x^{3} - 400 (3 x^{2})$

Step 1.1.4.3

Multiply $3$ by $- 400$ .

$f' (x) = - 60 x^{4} + 540 x^{3} - 1200 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 60 x^{4} + 540 x^{3} - 1200 x^{2}$ .

$- 60 x^{4} + 540 x^{3} - 1200 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 60 x^{4} + 540 x^{3} - 1200 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 60 x^{4} + 540 x^{3} - 1200 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $- 60 x^{2}$ out of $- 60 x^{4} + 540 x^{3} - 1200 x^{2}$ .

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Step 2.2.1.1

Factor $- 60 x^{2}$ out of $- 60 x^{4}$ .

$- 60 x^{2} x^{2} + 540 x^{3} - 1200 x^{2} = 0$

Step 2.2.1.2

Factor $- 60 x^{2}$ out of $540 x^{3}$ .

$- 60 x^{2} x^{2} - 60 x^{2} (- 9 x) - 1200 x^{2} = 0$

Step 2.2.1.3

Factor $- 60 x^{2}$ out of $- 1200 x^{2}$ .

$- 60 x^{2} x^{2} - 60 x^{2} (- 9 x) - 60 x^{2} \cdot 20 = 0$

Step 2.2.1.4

Factor $- 60 x^{2}$ out of $- 60 x^{2} (x^{2}) - 60 x^{2} (- 9 x)$ .

$- 60 x^{2} (x^{2} - 9 x) - 60 x^{2} \cdot 20 = 0$

Step 2.2.1.5

Factor $- 60 x^{2}$ out of $- 60 x^{2} (x^{2} - 9 x) - 60 x^{2} (20)$ .

$- 60 x^{2} (x^{2} - 9 x + 20) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} - 9 x + 20$ 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 $20$ and whose sum is $- 9$ .

$- 5, - 4$

Step 2.2.2.1.2

Write the factored form using these integers.

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

Step 2.2.2.2

Remove unnecessary parentheses.

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

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 $0$ .

$x^{2} = 0$

$x - 5 = 0$

$x - 4 = 0$

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

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Step 2.5.1

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

$x - 5 = 0$

Step 2.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.6

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.7

The final solution is all the values that make $- 60 x^{2} (x - 5) (x - 4) = 0$ true.

$x = 0, 5, 4$

Step 3

The values which make the derivative equal to $0$ are $0, 5, 4$ .

$0, 5, 4$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 4) \cup (4, 5) \cup (5, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

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

$f' (- 1) = - 60 {(- 1)}^{4} + 540 {(- 1)}^{3} - 1200 {(- 1)}^{2}$

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 $- 1$ to the power of $4$ .

$f' (- 1) = - 60 \cdot 1 + 540 {(- 1)}^{3} - 1200 {(- 1)}^{2}$

Step 5.2.1.2

Multiply $- 60$ by $1$ .

$f' (- 1) = - 60 + 540 {(- 1)}^{3} - 1200 {(- 1)}^{2}$

Step 5.2.1.3

Raise $- 1$ to the power of $3$ .

$f' (- 1) = - 60 + 540 \cdot - 1 - 1200 {(- 1)}^{2}$

Step 5.2.1.4

Multiply $540$ by $- 1$ .

$f' (- 1) = - 60 - 540 - 1200 {(- 1)}^{2}$

Step 5.2.1.5

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

$f' (- 1) = - 60 - 540 - 1200 \cdot 1$

Step 5.2.1.6

Multiply $- 1200$ by $1$ .

$f' (- 1) = - 60 - 540 - 1200$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $540$ from $- 60$ .

$f' (- 1) = - 600 - 1200$

Step 5.2.2.2

Subtract $1200$ from $- 600$ .

$f' (- 1) = - 1800$

Step 5.2.3

The final answer is $- 1800$ .

$- 1800$

Step 5.3

At $x = - 1$ the derivative is $- 1800$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

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

Step 6

Substitute a value from the interval $(0, 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

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

$f' (2) = - 60 {(2)}^{4} + 540 {(2)}^{3} - 1200 {(2)}^{2}$

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 $2$ to the power of $4$ .

$f' (2) = - 60 \cdot 16 + 540 {(2)}^{3} - 1200 {(2)}^{2}$

Step 6.2.1.2

Multiply $- 60$ by $16$ .

$f' (2) = - 960 + 540 {(2)}^{3} - 1200 {(2)}^{2}$

Step 6.2.1.3

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

$f' (2) = - 960 + 540 \cdot 8 - 1200 {(2)}^{2}$

Step 6.2.1.4

Multiply $540$ by $8$ .

$f' (2) = - 960 + 4320 - 1200 {(2)}^{2}$

Step 6.2.1.5

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

$f' (2) = - 960 + 4320 - 1200 \cdot 4$

Step 6.2.1.6

Multiply $- 1200$ by $4$ .

$f' (2) = - 960 + 4320 - 4800$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $- 960$ and $4320$ .

$f' (2) = 3360 - 4800$

Step 6.2.2.2

Subtract $4800$ from $3360$ .

$f' (2) = - 1440$

Step 6.2.3

The final answer is $- 1440$ .

$- 1440$

Step 6.3

At $x = 2$ the derivative is $- 1440$ . Since this is negative, the function is decreasing on $(0, 4)$ .

Decreasing on $(0, 4)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(4, 5)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $\frac{9}{2}$ in the expression.

$f' (\frac{9}{2}) = - 60 {(\frac{9}{2})}^{4} + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

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

Apply the product rule to $\frac{9}{2}$ .

$f' (\frac{9}{2}) = - 60 \frac{9^{4}}{2^{4}} + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.2

Raise $9$ to the power of $4$ .

$f' (\frac{9}{2}) = - 60 \frac{6561}{2^{4}} + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.3

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

$f' (\frac{9}{2}) = - 60 (\frac{6561}{16}) + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.4

Cancel the common factor of $4$ .

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Step 7.2.1.4.1

Factor $4$ out of $- 60$ .

$f' (\frac{9}{2}) = 4 (- 15) (\frac{6561}{16}) + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.4.2

Factor $4$ out of $16$ .

$f' (\frac{9}{2}) = 4 \cdot (- 15 \frac{6561}{4 \cdot 4}) + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.4.3

Cancel the common factor.

$f' (\frac{9}{2}) = 4 \cdot (- 15 \frac{6561}{4 \cdot 4}) + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.4.4

Rewrite the expression.

$f' (\frac{9}{2}) = - 15 (\frac{6561}{4}) + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.5

Combine $- 15$ and $\frac{6561}{4}$ .

$f' (\frac{9}{2}) = \frac{- 15 \cdot 6561}{4} + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.6

Multiply $- 15$ by $6561$ .

$f' (\frac{9}{2}) = \frac{- 98415}{4} + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.7

Move the negative in front of the fraction.

$f' (\frac{9}{2}) = - \frac{98415}{4} + 540 {(\frac{9}{2})}^{3} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.8

Apply the product rule to $\frac{9}{2}$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + 540 (\frac{9^{3}}{2^{3}}) - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.9

Raise $9$ to the power of $3$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + 540 (\frac{729}{2^{3}}) - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.10

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

$f' (\frac{9}{2}) = - \frac{98415}{4} + 540 (\frac{729}{8}) - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.11

Cancel the common factor of $4$ .

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Step 7.2.1.11.1

Factor $4$ out of $540$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + 4 (135) (\frac{729}{8}) - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.11.2

Factor $4$ out of $8$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + 4 \cdot (135 (\frac{729}{4 \cdot 2})) - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.11.3

Cancel the common factor.

$f' (\frac{9}{2}) = - \frac{98415}{4} + 4 \cdot (135 (\frac{729}{4 \cdot 2})) - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.11.4

Rewrite the expression.

$f' (\frac{9}{2}) = - \frac{98415}{4} + 135 (\frac{729}{2}) - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.12

Combine $135$ and $\frac{729}{2}$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{135 \cdot 729}{2} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.13

Multiply $135$ by $729$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} - 1200 {(\frac{9}{2})}^{2}$

Step 7.2.1.14

Apply the product rule to $\frac{9}{2}$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} - 1200 \frac{9^{2}}{2^{2}}$

Step 7.2.1.15

Raise $9$ to the power of $2$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} - 1200 \frac{81}{2^{2}}$

Step 7.2.1.16

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

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} - 1200 (\frac{81}{4})$

Step 7.2.1.17

Cancel the common factor of $4$ .

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Step 7.2.1.17.1

Factor $4$ out of $- 1200$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} + 4 (- 300) (\frac{81}{4})$

Step 7.2.1.17.2

Cancel the common factor.

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} + 4 \cdot (- 300 (\frac{81}{4}))$

Step 7.2.1.17.3

Rewrite the expression.

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} - 300 \cdot 81$

Step 7.2.1.18

Multiply $- 300$ by $81$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} - 24300$

Step 7.2.2

Find the common denominator.

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Step 7.2.2.1

Multiply $\frac{98415}{2}$ by $\frac{2}{2}$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415}{2} \cdot \frac{2}{2} - 24300$

Step 7.2.2.2

Multiply $\frac{98415}{2}$ by $\frac{2}{2}$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415 \cdot 2}{2 \cdot 2} - 24300$

Step 7.2.2.3

Write $- 24300$ as a fraction with denominator $1$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415 \cdot 2}{2 \cdot 2} + \frac{- 24300}{1}$

Step 7.2.2.4

Multiply $\frac{- 24300}{1}$ by $\frac{4}{4}$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415 \cdot 2}{2 \cdot 2} + \frac{- 24300}{1} \cdot \frac{4}{4}$

Step 7.2.2.5

Multiply $\frac{- 24300}{1}$ by $\frac{4}{4}$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415 \cdot 2}{2 \cdot 2} + \frac{- 24300 \cdot 4}{4}$

Step 7.2.2.6

Multiply $2$ by $2$ .

$f' (\frac{9}{2}) = - \frac{98415}{4} + \frac{98415 \cdot 2}{4} + \frac{- 24300 \cdot 4}{4}$

Step 7.2.3

Combine the numerators over the common denominator.

$f' (\frac{9}{2}) = \frac{- 98415 + 98415 \cdot 2 - 24300 \cdot 4}{4}$

Step 7.2.4

Simplify each term.

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Step 7.2.4.1

Multiply $98415$ by $2$ .

$f' (\frac{9}{2}) = \frac{- 98415 + 196830 - 24300 \cdot 4}{4}$

Step 7.2.4.2

Multiply $- 24300$ by $4$ .

$f' (\frac{9}{2}) = \frac{- 98415 + 196830 - 97200}{4}$

Step 7.2.5

Simplify by adding and subtracting.

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Step 7.2.5.1

Add $- 98415$ and $196830$ .

$f' (\frac{9}{2}) = \frac{98415 - 97200}{4}$

Step 7.2.5.2

Subtract $97200$ from $98415$ .

$f' (\frac{9}{2}) = \frac{1215}{4}$

Step 7.2.6

The final answer is $\frac{1215}{4}$ .

$\frac{1215}{4}$

Step 7.3

At $x = \frac{9}{2}$ the derivative is $\frac{1215}{4}$ . Since this is positive, the function is increasing on $(4, 5)$ .

Increasing on $(4, 5)$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(5, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

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

$f' (6) = - 60 {(6)}^{4} + 540 {(6)}^{3} - 1200 {(6)}^{2}$

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 $6$ to the power of $4$ .

$f' (6) = - 60 \cdot 1296 + 540 {(6)}^{3} - 1200 {(6)}^{2}$

Step 8.2.1.2

Multiply $- 60$ by $1296$ .

$f' (6) = - 77760 + 540 {(6)}^{3} - 1200 {(6)}^{2}$

Step 8.2.1.3

Raise $6$ to the power of $3$ .

$f' (6) = - 77760 + 540 \cdot 216 - 1200 {(6)}^{2}$

Step 8.2.1.4

Multiply $540$ by $216$ .

$f' (6) = - 77760 + 116640 - 1200 {(6)}^{2}$

Step 8.2.1.5

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

$f' (6) = - 77760 + 116640 - 1200 \cdot 36$

Step 8.2.1.6

Multiply $- 1200$ by $36$ .

$f' (6) = - 77760 + 116640 - 43200$

Step 8.2.2

Simplify by adding and subtracting.

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Step 8.2.2.1

Add $- 77760$ and $116640$ .

$f' (6) = 38880 - 43200$

Step 8.2.2.2

Subtract $43200$ from $38880$ .

$f' (6) = - 4320$

Step 8.2.3

The final answer is $- 4320$ .

$- 4320$

Step 8.3

At $x = 6$ the derivative is $- 4320$ . Since this is negative, the function is decreasing on $(5, \infty)$ .

Decreasing on $(5, \infty)$ since $f' (x) < 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Increasing on: $(4, 5)$

Decreasing on: $(- \infty, 0), (0, 4), (5, \infty)$

Step 10