Calculus Examples

$y = 8 x^{3} - 18 x^{2} - 24 x + 9$

Step 1

Write $y = 8 x^{3} - 18 x^{2} - 24 x + 9$ as a function.

$f (x) = 8 x^{3} - 18 x^{2} - 24 x + 9$

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 $8 x^{3} - 18 x^{2} - 24 x + 9$ with respect to $x$ is $\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$

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

$8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$

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 $n = 3$ .

$8 (3 x^{2}) + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$

Step 2.1.2.3

Multiply $3$ by $8$ .

$24 x^{2} + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 18 x^{2}]$ .

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Step 2.1.3.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x^{2}$ with respect to $x$ is $- 18 \frac{d}{d x} [x^{2}]$ .

$24 x^{2} - 18 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$

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 $n = 2$ .

$24 x^{2} - 18 (2 x) + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$

Step 2.1.3.3

Multiply $2$ by $- 18$ .

$24 x^{2} - 36 x + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [9]$

Step 2.1.4

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

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Step 2.1.4.1

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

$24 x^{2} - 36 x - 24 \frac{d}{d x} [x] + \frac{d}{d x} [9]$

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 $n = 1$ .

$24 x^{2} - 36 x - 24 \cdot 1 + \frac{d}{d x} [9]$

Step 2.1.4.3

Multiply $- 24$ by $1$ .

$24 x^{2} - 36 x - 24 + \frac{d}{d x} [9]$

Step 2.1.5

Differentiate using the Constant Rule.

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Step 2.1.5.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$24 x^{2} - 36 x - 24 + 0$

Step 2.1.5.2

Add $24 x^{2} - 36 x - 24$ and $0$ .

$f' (x) = 24 x^{2} - 36 x - 24$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $24 x^{2} - 36 x - 24$ .

$24 x^{2} - 36 x - 24$

Step 3

Set the first derivative equal to $0$ then solve the equation $24 x^{2} - 36 x - 24 = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$24 x^{2} - 36 x - 24 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $12$ out of $24 x^{2} - 36 x - 24$ .

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Step 3.2.1.1

Factor $12$ out of $24 x^{2}$ .

$12 (2 x^{2}) - 36 x - 24 = 0$

Step 3.2.1.2

Factor $12$ out of $- 36 x$ .

$12 (2 x^{2}) + 12 (- 3 x) - 24 = 0$

Step 3.2.1.3

Factor $12$ out of $- 24$ .

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

Step 3.2.1.4

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

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

Step 3.2.1.5

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

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

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

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Step 3.2.2.1.1.1

Factor $- 3$ out of $- 3 x$ .

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

Step 3.2.2.1.1.2

Rewrite $- 3$ as $1$ plus $- 4$

$12 (2 x^{2} + (1 - 4) x - 2) = 0$

Step 3.2.2.1.1.3

Apply the distributive property.

$12 (2 x^{2} + 1 x - 4 x - 2) = 0$

Step 3.2.2.1.1.4

Multiply $x$ by $1$ .

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

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.

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

Step 3.2.2.1.2.2

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

$12 (x (2 x + 1) - 2 (2 x + 1)) = 0$

Step 3.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$12 ((2 x + 1) (x - 2)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$12 (2 x + 1) (x - 2) = 0$

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 $0$ .

$2 x + 1 = 0$

$x - 2 = 0$

Step 3.4

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 3.4.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 3.4.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 3.4.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 3.4.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

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.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

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.

$x = - \frac{1}{2}$

Step 3.5

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

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Step 3.5.1

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

$x - 2 = 0$

Step 3.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.6

The final solution is all the values that make $12 (2 x + 1) (x - 2) = 0$ true.

$x = - \frac{1}{2}, 2$

Step 4

The values which make the derivative equal to $0$ are $- \frac{1}{2}, 2$ .

$- \frac{1}{2}, 2$

Step 5

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

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, 2) \cup (2, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - \frac{1}{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.5$ in the expression.

$f' (- 1.5) = 24 {(- 1.5)}^{2} - 36 \cdot - 1.5 - 24$

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 $- 1.5$ to the power of $2$ .

$f' (- 1.5) = 24 \cdot 2.25 - 36 \cdot - 1.5 - 24$

Step 6.2.1.2

Multiply $24$ by $2.25$ .

$f' (- 1.5) = 54 - 36 \cdot - 1.5 - 24$

Step 6.2.1.3

Multiply $- 36$ by $- 1.5$ .

$f' (- 1.5) = 54 + 54 - 24$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $54$ and $54$ .

$f' (- 1.5) = 108 - 24$

Step 6.2.2.2

Subtract $24$ from $108$ .

$f' (- 1.5) = 84$

Step 6.2.3

The final answer is $84$ .

$84$

Step 6.3

At $x = - 1.5$ the derivative is $84$ . Since this is positive, the function is increasing on $(- \infty, - \frac{1}{2})$ .

Increasing on $(- \infty, - \frac{1}{2})$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 0.5, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.75$ in the expression.

$f' (0.75) = 24 {(0.75)}^{2} - 36 \cdot 0.75 - 24$

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 $0.75$ to the power of $2$ .

$f' (0.75) = 24 \cdot 0.5625 - 36 \cdot 0.75 - 24$

Step 7.2.1.2

Multiply $24$ by $0.5625$ .

$f' (0.75) = 13.5 - 36 \cdot 0.75 - 24$

Step 7.2.1.3

Multiply $- 36$ by $0.75$ .

$f' (0.75) = 13.5 - 27 - 24$

Step 7.2.2

Simplify by subtracting numbers.

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Step 7.2.2.1

Subtract $27$ from $13.5$ .

$f' (0.75) = - 13.5 - 24$

Step 7.2.2.2

Subtract $24$ from $- 13.5$ .

$f' (0.75) = - 37.5$

Step 7.2.3

The final answer is $- 37.5$ .

$- 37.5$

Step 7.3

At $x = 0.75$ the derivative is $- 37.5$ . Since this is negative, the function is decreasing on $(- 0.5, 2)$ .

Decreasing on $(- \frac{1}{2}, 2)$ since $f' (x) < 0$

Step 8

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

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Step 8.1

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

$f' (3) = 24 {(3)}^{2} - 36 \cdot 3 - 24$

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 $3$ to the power of $2$ .

$f' (3) = 24 \cdot 9 - 36 \cdot 3 - 24$

Step 8.2.1.2

Multiply $24$ by $9$ .

$f' (3) = 216 - 36 \cdot 3 - 24$

Step 8.2.1.3

Multiply $- 36$ by $3$ .

$f' (3) = 216 - 108 - 24$

Step 8.2.2

Simplify by subtracting numbers.

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Step 8.2.2.1

Subtract $108$ from $216$ .

$f' (3) = 108 - 24$

Step 8.2.2.2

Subtract $24$ from $108$ .

$f' (3) = 84$

Step 8.2.3

The final answer is $84$ .

$84$

Step 8.3

At $x = 3$ the derivative is $84$ . Since this is positive, the function is increasing on $(2, \infty)$ .

Increasing on $(2, \infty)$ since $f' (x) > 0$

Step 9

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

Increasing on: $(- \infty, - \frac{1}{2}), (2, \infty)$

Decreasing on: $(- \frac{1}{2}, 2)$

Step 10