Calculus Examples

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

Step 1

Write $y = 12 x^{2} - 102 x - 180$ as a function.

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

Step 2

Find the first derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $12 x^{2} - 102 x - 180$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 102 x] + \frac{d}{d x} [- 180]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 102 x] + \frac{d}{d x} [- 180]$

Step 2.1.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

Tap for more steps...

Step 2.1.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 102 x] + \frac{d}{d x} [- 180]$

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 $n = 2$ .

$12 (2 x) + \frac{d}{d x} [- 102 x] + \frac{d}{d x} [- 180]$

Step 2.1.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [- 102 x] + \frac{d}{d x} [- 180]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 102 x]$ .

Tap for more steps...

Step 2.1.3.1

Since $- 102$ is constant with respect to $x$ , the derivative of $- 102 x$ with respect to $x$ is $- 102 \frac{d}{d x} [x]$ .

$24 x - 102 \frac{d}{d x} [x] + \frac{d}{d x} [- 180]$

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 $n = 1$ .

$24 x - 102 \cdot 1 + \frac{d}{d x} [- 180]$

Step 2.1.3.3

Multiply $- 102$ by $1$ .

$24 x - 102 + \frac{d}{d x} [- 180]$

Step 2.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.4.1

Since $- 180$ is constant with respect to $x$ , the derivative of $- 180$ with respect to $x$ is $0$ .

$24 x - 102 + 0$

Step 2.1.4.2

Add $24 x - 102$ and $0$ .

$f' (x) = 24 x - 102$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $24 x - 102$ .

$24 x - 102$

Step 3

Set the first derivative equal to $0$ then solve the equation $24 x - 102 = 0$ .

Tap for more steps...

Step 3.1

Set the first derivative equal to $0$ .

$24 x - 102 = 0$

Step 3.2

Add $102$ to both sides of the equation.

$24 x = 102$

Step 3.3

Divide each term in $24 x = 102$ by $24$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $24 x = 102$ by $24$ .

$\frac{24 x}{24} = \frac{102}{24}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $24$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{102}{24}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{102}{24}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Cancel the common factor of $102$ and $24$ .

Tap for more steps...

Step 3.3.3.1.1

Factor $6$ out of $102$ .

$x = \frac{6 (17)}{24}$

Step 3.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.2.1

Factor $6$ out of $24$ .

$x = \frac{6 \cdot 17}{6 \cdot 4}$

Step 3.3.3.1.2.2

Cancel the common factor.

$x = \frac{6 \cdot 17}{6 \cdot 4}$

Step 3.3.3.1.2.3

Rewrite the expression.

$x = \frac{17}{4}$

Step 4

The values which make the derivative equal to $0$ are $\frac{17}{4}$ .

$\frac{17}{4}$

Step 5

After finding the point that makes the derivative $f' (x) = 24 x - 102$ equal to $0$ or undefined, the interval to check where $f (x) = 12 x^{2} - 102 x - 180$ is increasing and where it is decreasing is $(- \infty, \frac{17}{4}) \cup (\frac{17}{4}, \infty)$ .

$(- \infty, \frac{17}{4}) \cup (\frac{17}{4}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, \frac{17}{4})$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $\frac{13}{4}$ in the expression.

$f' (\frac{13}{4}) = 24 (\frac{13}{4}) - 102$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.1.1.1

Factor $4$ out of $24$ .

$f' (\frac{13}{4}) = 4 (6) (\frac{13}{4}) - 102$

Step 6.2.1.1.2

Cancel the common factor.

$f' (\frac{13}{4}) = 4 \cdot (6 (\frac{13}{4})) - 102$

Step 6.2.1.1.3

Rewrite the expression.

$f' (\frac{13}{4}) = 6 \cdot 13 - 102$

Step 6.2.1.2

Multiply $6$ by $13$ .

$f' (\frac{13}{4}) = 78 - 102$

Step 6.2.2

Subtract $102$ from $78$ .

$f' (\frac{13}{4}) = - 24$

Step 6.2.3

The final answer is $- 24$ .

$- 24$

Step 6.3

At $x = \frac{13}{4}$ the derivative is $- 24$ . Since this is negative, the function is decreasing on $(- \infty, \frac{17}{4})$ .

Decreasing on $(- \infty, \frac{17}{4})$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{17}{4}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $\frac{21}{4}$ in the expression.

$f' (\frac{21}{4}) = 24 (\frac{21}{4}) - 102$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.2.1.1.1

Factor $4$ out of $24$ .

$f' (\frac{21}{4}) = 4 (6) (\frac{21}{4}) - 102$

Step 7.2.1.1.2

Cancel the common factor.

$f' (\frac{21}{4}) = 4 \cdot (6 (\frac{21}{4})) - 102$

Step 7.2.1.1.3

Rewrite the expression.

$f' (\frac{21}{4}) = 6 \cdot 21 - 102$

Step 7.2.1.2

Multiply $6$ by $21$ .

$f' (\frac{21}{4}) = 126 - 102$

Step 7.2.2

Subtract $102$ from $126$ .

$f' (\frac{21}{4}) = 24$

Step 7.2.3

The final answer is $24$ .

$24$

Step 7.3

At $x = \frac{21}{4}$ the derivative is $24$ . Since this is positive, the function is increasing on $(\frac{17}{4}, \infty)$ .

Increasing on $(\frac{17}{4}, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(\frac{17}{4}, \infty)$

Decreasing on: $(- \infty, \frac{17}{4})$

Step 9