Calculus Examples

$f (x) = x^{3} - 6 x^{2} + 9 x$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $x^{3} - 6 x^{2} + 9 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [9 x]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (9 x)$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 3 x^{2} + \frac{d}{d x} (- 6 x^{2}) + \frac{d}{d x} (9 x)$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

Tap for more steps...

Step 1.1.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 3 x^{2} - 6 \frac{d}{d x} x^{2} + \frac{d}{d x} (9 x)$

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 $n = 2$ .

$f' (x) = 3 x^{2} - 6 (2 x) + \frac{d}{d x} (9 x)$

Step 1.1.2.3

Multiply $2$ by $- 6$ .

$f' (x) = 3 x^{2} - 12 x + \frac{d}{d x} (9 x)$

Step 1.1.3

Evaluate $\frac{d}{d x} [9 x]$ .

Tap for more steps...

Step 1.1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} - 12 x + 9 \frac{d}{d x} (x)$

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 $n = 1$ .

$f' (x) = 3 x^{2} - 12 x + 9 \cdot 1$

Step 1.1.3.3

Multiply $9$ by $1$ .

$f' (x) = 3 x^{2} - 12 x + 9$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 12 x + 9$ .

$3 x^{2} - 12 x + 9$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 12 x + 9 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 12 x + 9 = 0$

Step 2.2

Factor the left side of the equation.

Tap for more steps...

Step 2.2.1

Factor $3$ out of $3 x^{2} - 12 x + 9$ .

Tap for more steps...

Step 2.2.1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) - 12 x + 9 = 0$

Step 2.2.1.2

Factor $3$ out of $- 12 x$ .

$3 (x^{2}) + 3 (- 4 x) + 9 = 0$

Step 2.2.1.3

Factor $3$ out of $9$ .

$3 x^{2} + 3 (- 4 x) + 3 \cdot 3 = 0$

Step 2.2.1.4

Factor $3$ out of $3 x^{2} + 3 (- 4 x)$ .

$3 (x^{2} - 4 x) + 3 \cdot 3 = 0$

Step 2.2.1.5

Factor $3$ out of $3 (x^{2} - 4 x) + 3 \cdot 3$ .

$3 (x^{2} - 4 x + 3) = 0$

Step 2.2.2

Factor.

Tap for more steps...

Step 2.2.2.1

Factor $x^{2} - 4 x + 3$ using the AC method.

Tap for more steps...

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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 2.2.2.1.2

Write the factored form using these integers.

$3 ((x - 3) (x - 1)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$3 (x - 3) (x - 1) = 0$

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 $0$ .

$x - 3 = 0$

$x - 1 = 0$

Step 2.4

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.6

The final solution is all the values that make $3 (x - 3) (x - 1) = 0$ true.

$x = 3, 1$

Step 3

The values which make the derivative equal to $0$ are $3, 1$ .

$3, 1$

Step 4

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

$(- \infty, 1) \cup (1, 3) \cup (3, \infty)$

Step 5

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

Tap for more steps...

Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = 3 {(0)}^{2} - 12 \cdot 0 + 9$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Raising $0$ to any positive power yields $0$ .

$f' (0) = 3 \cdot 0 - 12 \cdot 0 + 9$

Step 5.2.1.2

Multiply $3$ by $0$ .

$f' (0) = 0 - 12 \cdot 0 + 9$

Step 5.2.1.3

Multiply $- 12$ by $0$ .

$f' (0) = 0 + 0 + 9$

Step 5.2.2

Simplify by adding numbers.

Tap for more steps...

Step 5.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 + 9$

Step 5.2.2.2

Add $0$ and $9$ .

$f' (0) = 9$

Step 5.2.3

The final answer is $9$ .

$9$

Step 5.3

At $x = 0$ the derivative is $9$ . Since this is positive, the function is increasing on $(- \infty, 1)$ .

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

Step 6

Substitute a value from the interval $(1, 3)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

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

$f' (2) = 3 {(2)}^{2} - 12 \cdot 2 + 9$

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

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

$f' (2) = 3 \cdot 4 - 12 \cdot 2 + 9$

Step 6.2.1.2

Multiply $3$ by $4$ .

$f' (2) = 12 - 12 \cdot 2 + 9$

Step 6.2.1.3

Multiply $- 12$ by $2$ .

$f' (2) = 12 - 24 + 9$

Step 6.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 6.2.2.1

Subtract $24$ from $12$ .

$f' (2) = - 12 + 9$

Step 6.2.2.2

Add $- 12$ and $9$ .

$f' (2) = - 3$

Step 6.2.3

The final answer is $- 3$ .

$- 3$

Step 6.3

At $x = 2$ the derivative is $- 3$ . Since this is negative, the function is decreasing on $(1, 3)$ .

Decreasing on $(1, 3)$ since $f' (x) < 0$

Step 7

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

Tap for more steps...

Step 7.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = 3 {(4)}^{2} - 12 \cdot 4 + 9$

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

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

$f' (4) = 3 \cdot 16 - 12 \cdot 4 + 9$

Step 7.2.1.2

Multiply $3$ by $16$ .

$f' (4) = 48 - 12 \cdot 4 + 9$

Step 7.2.1.3

Multiply $- 12$ by $4$ .

$f' (4) = 48 - 48 + 9$

Step 7.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 7.2.2.1

Subtract $48$ from $48$ .

$f' (4) = 0 + 9$

Step 7.2.2.2

Add $0$ and $9$ .

$f' (4) = 9$

Step 7.2.3

The final answer is $9$ .

$9$

Step 7.3

At $x = 4$ the derivative is $9$ . Since this is positive, the function is increasing on $(3, \infty)$ .

Increasing on $(3, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(- \infty, 1), (3, \infty)$

Decreasing on: $(1, 3)$

Step 9