Calculus Examples

$f (x) = 3 x^{2} - 12 x - 8$

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 $3 x^{2} - 12 x - 8$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 8]$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 8]$

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 $n = 2$ .

$3 (2 x) + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 8]$

Step 1.1.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [- 8]$

Step 1.1.3

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

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Step 1.1.3.1

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

$6 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [- 8]$

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 $n = 1$ .

$6 x - 12 \cdot 1 + \frac{d}{d x} [- 8]$

Step 1.1.3.3

Multiply $- 12$ by $1$ .

$6 x - 12 + \frac{d}{d x} [- 8]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$6 x - 12 + 0$

Step 1.1.4.2

Add $6 x - 12$ and $0$ .

$f' (x) = 6 x - 12$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $6 x - 12$ .

$6 x - 12$

Step 2

Set the first derivative equal to $0$ then solve the equation $6 x - 12 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$6 x - 12 = 0$

Step 2.2

Add $12$ to both sides of the equation.

$6 x = 12$

Step 2.3

Divide each term in $6 x = 12$ by $6$ and simplify.

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Step 2.3.1

Divide each term in $6 x = 12$ by $6$ .

$\frac{6 x}{6} = \frac{12}{6}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $6$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{12}{6}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{6}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $12$ by $6$ .

$x = 2$

Step 3

The values which make the derivative equal to $0$ are $2$ .

$2$

Step 4

After finding the point that makes the derivative $f' (x) = 6 x - 12$ equal to $0$ or undefined, the interval to check where $f (x) = 3 x^{2} - 12 x - 8$ is increasing and where it is decreasing is $(- \infty, 2) \cup (2, \infty)$ .

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

Step 5

Substitute a value from the interval $(- \infty, 2)$ 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.

$f' (1) = 6 (1) - 12$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $6$ by $1$ .

$f' (1) = 6 - 12$

Step 5.2.2

Subtract $12$ from $6$ .

$f' (1) = - 6$

Step 5.2.3

The final answer is $- 6$ .

$- 6$

Step 5.3

At $x = 1$ the derivative is $- 6$ . Since this is negative, the function is decreasing on $(- \infty, 2)$ .

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

Step 6

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 6.1

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

$f' (3) = 6 (3) - 12$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $6$ by $3$ .

$f' (3) = 18 - 12$

Step 6.2.2

Subtract $12$ from $18$ .

$f' (3) = 6$

Step 6.2.3

The final answer is $6$ .

$6$

Step 6.3

At $x = 3$ the derivative is $6$ . Since this is positive, the function is increasing on $(2, \infty)$ .

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

Step 7

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

Increasing on: $(2, \infty)$

Decreasing on: $(- \infty, 2)$

Step 8