Calculus Examples

$y = 4 x^{3} - 51 x^{2} - 180 x + 11$

Step 1

Write $y = 4 x^{3} - 51 x^{2} - 180 x + 11$ as a function.

$f (x) = 4 x^{3} - 51 x^{2} - 180 x + 11$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $4 x^{3} - 51 x^{2} - 180 x + 11$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 51 x^{2}] + \frac{d}{d x} [- 180 x] + \frac{d}{d x} [11]$ .

$f' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 51 x^{2}) + \frac{d}{d x} (- 180 x) + \frac{d}{d x} (11)$

Step 2.1.2

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

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Step 2.1.2.1

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

$f' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 51 x^{2}) + \frac{d}{d x} (- 180 x) + \frac{d}{d x} (11)$

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 $n = 3$ .

$f' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 51 x^{2}) + \frac{d}{d x} (- 180 x) + \frac{d}{d x} (11)$

Step 2.1.2.3

Multiply $3$ by $4$ .

$f' (x) = 12 x^{2} + \frac{d}{d x} (- 51 x^{2}) + \frac{d}{d x} (- 180 x) + \frac{d}{d x} (11)$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 51 x^{2}]$ .

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Step 2.1.3.1

Since $- 51$ is constant with respect to $x$ , the derivative of $- 51 x^{2}$ with respect to $x$ is $- 51 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 12 x^{2} - 51 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 180 x) + \frac{d}{d x} (11)$

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 $n = 2$ .

$f' (x) = 12 x^{2} - 51 (2 x) + \frac{d}{d x} (- 180 x) + \frac{d}{d x} (11)$

Step 2.1.3.3

Multiply $2$ by $- 51$ .

$f' (x) = 12 x^{2} - 102 x + \frac{d}{d x} (- 180 x) + \frac{d}{d x} (11)$

Step 2.1.4

Evaluate $\frac{d}{d x} [- 180 x]$ .

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Step 2.1.4.1

Since $- 180$ is constant with respect to $x$ , the derivative of $- 180 x$ with respect to $x$ is $- 180 \frac{d}{d x} [x]$ .

$f' (x) = 12 x^{2} - 102 x - 180 \frac{d}{d x} x + \frac{d}{d x} (11)$

Step 2.1.4.2

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

$f' (x) = 12 x^{2} - 102 x - 180 \cdot 1 + \frac{d}{d x} (11)$

Step 2.1.4.3

Multiply $- 180$ by $1$ .

$f' (x) = 12 x^{2} - 102 x - 180 + \frac{d}{d x} (11)$

Step 2.1.5

Differentiate using the Constant Rule.

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Step 2.1.5.1

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$f' (x) = 12 x^{2} - 102 x - 180 + 0$

Step 2.1.5.2

Add $12 x^{2} - 102 x - 180$ and $0$ .

$f' (x) = 12 x^{2} - 102 x - 180$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 102 x - 180$ .

$12 x^{2} - 102 x - 180$

Step 3

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 102 x - 180 = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$12 x^{2} - 102 x - 180 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $6$ out of $12 x^{2} - 102 x - 180$ .

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Step 3.2.1.1

Factor $6$ out of $12 x^{2}$ .

$6 (2 x^{2}) - 102 x - 180 = 0$

Step 3.2.1.2

Factor $6$ out of $- 102 x$ .

$6 (2 x^{2}) + 6 (- 17 x) - 180 = 0$

Step 3.2.1.3

Factor $6$ out of $- 180$ .

$6 (2 x^{2}) + 6 (- 17 x) + 6 (- 30) = 0$

Step 3.2.1.4

Factor $6$ out of $6 (2 x^{2}) + 6 (- 17 x)$ .

$6 (2 x^{2} - 17 x) + 6 (- 30) = 0$

Step 3.2.1.5

Factor $6$ out of $6 (2 x^{2} - 17 x) + 6 (- 30)$ .

$6 (2 x^{2} - 17 x - 30) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor by grouping.

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Step 3.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 30 = - 60$ and whose sum is $b = - 17$ .

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Step 3.2.2.1.1.1

Factor $- 17$ out of $- 17 x$ .

$6 (2 x^{2} - 17 x - 30) = 0$

Step 3.2.2.1.1.2

Rewrite $- 17$ as $3$ plus $- 20$

$6 (2 x^{2} + (3 - 20) x - 30) = 0$

Step 3.2.2.1.1.3

Apply the distributive property.

$6 (2 x^{2} + 3 x - 20 x - 30) = 0$

Step 3.2.2.1.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1.2.1

Group the first two terms and the last two terms.

$6 ((2 x^{2} + 3 x) - 20 x - 30) = 0$

Step 3.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$6 (x (2 x + 3) - 10 (2 x + 3)) = 0$

Step 3.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$6 ((2 x + 3) (x - 10)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$6 (2 x + 3) (x - 10) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 3 = 0$

$x - 10 = 0$

Step 3.4

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 3.4.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 3.4.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 3.4.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 3.4.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

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

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.4.2.2.3

Simplify the right side.

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Step 3.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 3.5

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 3.5.2

Add $10$ to both sides of the equation.

$x = 10$

Step 3.6

The final solution is all the values that make $6 (2 x + 3) (x - 10) = 0$ true.

$x = - \frac{3}{2}, 10$

Step 4

The values which make the derivative equal to $0$ are $- \frac{3}{2}, 10$ .

$- \frac{3}{2}, 10$

Step 5

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

$(- \infty, - \frac{3}{2}) \cup (- \frac{3}{2}, 10) \cup (10, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - \frac{3}{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 2.5$ in the expression.

$f' (- 2.5) = 12 {(- 2.5)}^{2} - 102 \cdot - 2.5 - 180$

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 $- 2.5$ to the power of $2$ .

$f' (- 2.5) = 12 \cdot 6.25 - 102 \cdot - 2.5 - 180$

Step 6.2.1.2

Multiply $12$ by $6.25$ .

$f' (- 2.5) = 75 - 102 \cdot - 2.5 - 180$

Step 6.2.1.3

Multiply $- 102$ by $- 2.5$ .

$f' (- 2.5) = 75 + 255 - 180$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $75$ and $255$ .

$f' (- 2.5) = 330 - 180$

Step 6.2.2.2

Subtract $180$ from $330$ .

$f' (- 2.5) = 150$

Step 6.2.3

The final answer is $150$ .

$150$

Step 6.3

At $x = - 2.5$ the derivative is $150$ . Since this is positive, the function is increasing on $(- \infty, - \frac{3}{2})$ .

Increasing on $(- \infty, - \frac{3}{2})$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 1.5, 10)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $4.25$ in the expression.

$f' (4.25) = 12 {(4.25)}^{2} - 102 \cdot 4.25 - 180$

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 $4.25$ to the power of $2$ .

$f' (4.25) = 12 \cdot 18.0625 - 102 \cdot 4.25 - 180$

Step 7.2.1.2

Multiply $12$ by $18.0625$ .

$f' (4.25) = 216.75 - 102 \cdot 4.25 - 180$

Step 7.2.1.3

Multiply $- 102$ by $4.25$ .

$f' (4.25) = 216.75 - 433.5 - 180$

Step 7.2.2

Simplify by subtracting numbers.

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Step 7.2.2.1

Subtract $433.5$ from $216.75$ .

$f' (4.25) = - 216.75 - 180$

Step 7.2.2.2

Subtract $180$ from $- 216.75$ .

$f' (4.25) = - 396.75$

Step 7.2.3

The final answer is $- 396.75$ .

$- 396.75$

Step 7.3

At $x = 4.25$ the derivative is $- 396.75$ . Since this is negative, the function is decreasing on $(- 1.5, 10)$ .

Decreasing on $(- \frac{3}{2}, 10)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(10, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $11$ in the expression.

$f' (11) = 12 {(11)}^{2} - 102 \cdot 11 - 180$

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 $11$ to the power of $2$ .

$f' (11) = 12 \cdot 121 - 102 \cdot 11 - 180$

Step 8.2.1.2

Multiply $12$ by $121$ .

$f' (11) = 1452 - 102 \cdot 11 - 180$

Step 8.2.1.3

Multiply $- 102$ by $11$ .

$f' (11) = 1452 - 1122 - 180$

Step 8.2.2

Simplify by subtracting numbers.

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Step 8.2.2.1

Subtract $1122$ from $1452$ .

$f' (11) = 330 - 180$

Step 8.2.2.2

Subtract $180$ from $330$ .

$f' (11) = 150$

Step 8.2.3

The final answer is $150$ .

$150$

Step 8.3

At $x = 11$ the derivative is $150$ . Since this is positive, the function is increasing on $(10, \infty)$ .

Increasing on $(10, \infty)$ since $f' (x) > 0$

Step 9

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

Increasing on: $(- \infty, - \frac{3}{2}), (10, \infty)$

Decreasing on: $(- \frac{3}{2}, 10)$

Step 10