Calculus Examples

$f (x) = - 3 x^{4} - 8 x^{3} - 6 x^{2} + 1$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $- 3 x^{4} - 8 x^{3} - 6 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$- 3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 3 (4 x^{3}) + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.2.3

Multiply $4$ by $- 3$ .

$- 12 x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.3

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

Tap for more steps...

Step 1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$- 12 x^{3} - 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.3.2

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

$- 12 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.3.3

Multiply $3$ by $- 8$ .

$- 12 x^{3} - 24 x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

$- 12 x^{3} - 24 x^{2} - 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 12 x^{3} - 24 x^{2} - 6 (2 x) + \frac{d}{d x} [1]$

Step 1.4.3

Multiply $2$ by $- 6$ .

$- 12 x^{3} - 24 x^{2} - 12 x + \frac{d}{d x} [1]$

Step 1.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.5.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.5.2

Add $- 12 x^{3} - 24 x^{2} - 12 x$ and $0$ .

$- 12 x^{3} - 24 x^{2} - 12 x$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- 12 x^{3} - 24 x^{2} - 12 x$ with respect to $x$ is $\frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [- 12 x]$ .

$f'' (x) = \frac{d}{d x} (- 12 x^{3}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (- 12 x)$

Step 2.2

Evaluate $\frac{d}{d x} [- 12 x^{3}]$ .

Tap for more steps...

Step 2.2.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{3}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = - 12 \frac{d}{d x} x^{3} + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (- 12 x)$

Step 2.2.2

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

$f'' (x) = - 12 (3 x^{2}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (- 12 x)$

Step 2.2.3

Multiply $3$ by $- 12$ .

$f'' (x) = - 36 x^{2} + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (- 12 x)$

Step 2.3

Evaluate $\frac{d}{d x} [- 24 x^{2}]$ .

Tap for more steps...

Step 2.3.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x^{2}$ with respect to $x$ is $- 24 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - 36 x^{2} - 24 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 12 x)$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = - 36 x^{2} - 24 (2 x) + \frac{d}{d x} (- 12 x)$

Step 2.3.3

Multiply $2$ by $- 24$ .

$f'' (x) = - 36 x^{2} - 48 x + \frac{d}{d x} (- 12 x)$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$f'' (x) = - 36 x^{2} - 48 x - 12 \frac{d}{d x} x$

Step 2.4.2

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

$f'' (x) = - 36 x^{2} - 48 x - 12 \cdot 1$

Step 2.4.3

Multiply $- 12$ by $1$ .

$f'' (x) = - 36 x^{2} - 48 x - 12$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

By the Sum Rule, the derivative of $- 3 x^{4} - 8 x^{3} - 6 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

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

$- 3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 3 (4 x^{3}) + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 4.1.2.3

Multiply $4$ by $- 3$ .

$- 12 x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 4.1.3

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

Tap for more steps...

Step 4.1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$- 12 x^{3} - 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 4.1.3.2

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

$- 12 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 4.1.3.3

Multiply $3$ by $- 8$ .

$- 12 x^{3} - 24 x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [1]$

Step 4.1.4

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

Tap for more steps...

Step 4.1.4.1

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

$- 12 x^{3} - 24 x^{2} - 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 4.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 12 x^{3} - 24 x^{2} - 6 (2 x) + \frac{d}{d x} [1]$

Step 4.1.4.3

Multiply $2$ by $- 6$ .

$- 12 x^{3} - 24 x^{2} - 12 x + \frac{d}{d x} [1]$

Step 4.1.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 4.1.5.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 4.1.5.2

Add $- 12 x^{3} - 24 x^{2} - 12 x$ and $0$ .

$f' (x) = - 12 x^{3} - 24 x^{2} - 12 x$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 12 x^{3} - 24 x^{2} - 12 x$ .

$- 12 x^{3} - 24 x^{2} - 12 x$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 12 x^{3} - 24 x^{2} - 12 x = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Factor the left side of the equation.

Tap for more steps...

Step 5.2.1

Factor $- 12 x$ out of $- 12 x^{3} - 24 x^{2} - 12 x$ .

Tap for more steps...

Step 5.2.1.1

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

$- 12 x \cdot x^{2} - 24 x^{2} - 12 x = 0$

Step 5.2.1.2

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

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

Step 5.2.1.3

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

$- 12 x \cdot x^{2} - 12 x (2 x) - 12 x \cdot 1 = 0$

Step 5.2.1.4

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

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

Step 5.2.1.5

Factor $- 12 x$ out of $- 12 x (x^{2} + 2 x) - 12 x (1)$ .

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

Step 5.2.2

Factor using the perfect square rule.

Tap for more steps...

Step 5.2.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 5.2.2.3

Rewrite the polynomial.

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

Step 5.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 5.3

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

$x = 0$

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

Step 5.4

Set $x$ equal to $0$ .

$x = 0$

Step 5.5

Set ${(x + 1)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.1

Set ${(x + 1)}^{2}$ equal to $0$ .

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

Step 5.5.2

Solve ${(x + 1)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 5.5.2.1

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

$x + 1 = 0$

Step 5.5.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5.6

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

$x = 0, - 1$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0, - 1$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 36 {(0)}^{2} - 48 \cdot 0 - 12$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

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

$- 36 \cdot 0 - 48 \cdot 0 - 12$

Step 9.1.2

Multiply $- 36$ by $0$ .

$0 - 48 \cdot 0 - 12$

Step 9.1.3

Multiply $- 48$ by $0$ .

$0 + 0 - 12$

Step 9.2

Simplify by adding and subtracting.

Tap for more steps...

Step 9.2.1

Add $0$ and $0$ .

$0 - 12$

Step 9.2.2

Subtract $12$ from $0$ .

$- 12$

Step 10

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 11

Find the y-value when $x = 0$ .

Tap for more steps...

Step 11.1

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

$f (0) = - 3 {(0)}^{4} - 8 {(0)}^{3} - 6 {(0)}^{2} + 1$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1

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

$f (0) = - 3 \cdot 0 - 8 {(0)}^{3} - 6 {(0)}^{2} + 1$

Step 11.2.1.2

Multiply $- 3$ by $0$ .

$f (0) = 0 - 8 {(0)}^{3} - 6 {(0)}^{2} + 1$

Step 11.2.1.3

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

$f (0) = 0 - 8 \cdot 0 - 6 {(0)}^{2} + 1$

Step 11.2.1.4

Multiply $- 8$ by $0$ .

$f (0) = 0 + 0 - 6 {(0)}^{2} + 1$

Step 11.2.1.5

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

$f (0) = 0 + 0 - 6 \cdot 0 + 1$

Step 11.2.1.6

Multiply $- 6$ by $0$ .

$f (0) = 0 + 0 + 0 + 1$

Step 11.2.2

Simplify by adding numbers.

Tap for more steps...

Step 11.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 1$

Step 11.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 1$

Step 11.2.2.3

Add $0$ and $1$ .

$f (0) = 1$

Step 11.2.3

The final answer is $1$ .

$y = 1$

Step 12

Evaluate the second derivative at $x = - 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 36 {(- 1)}^{2} - 48 \cdot - 1 - 12$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

Raise $- 1$ to the power of $2$ .

$- 36 \cdot 1 - 48 \cdot - 1 - 12$

Step 13.1.2

Multiply $- 36$ by $1$ .

$- 36 - 48 \cdot - 1 - 12$

Step 13.1.3

Multiply $- 48$ by $- 1$ .

$- 36 + 48 - 12$

Step 13.2

Simplify by adding and subtracting.

Tap for more steps...

Step 13.2.1

Add $- 36$ and $48$ .

$12 - 12$

Step 13.2.2

Subtract $12$ from $12$ .

$0$

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Tap for more steps...

Step 14.1

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

$(- \infty, - 1) \cup (- 1, 0) \cup (0, \infty)$

Step 14.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - 1)$ in the first derivative $- 12 x^{3} - 24 x^{2} - 12 x$ to check if the result is negative or positive.

Tap for more steps...

Step 14.2.1

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

$f' (- 4) = - 12 {(- 4)}^{3} - 24 {(- 4)}^{2} - 12 \cdot - 4$

Step 14.2.2

Simplify the result.

Tap for more steps...

Step 14.2.2.1

Simplify each term.

Tap for more steps...

Step 14.2.2.1.1

Raise $- 4$ to the power of $3$ .

$f' (- 4) = - 12 \cdot - 64 - 24 {(- 4)}^{2} - 12 \cdot - 4$

Step 14.2.2.1.2

Multiply $- 12$ by $- 64$ .

$f' (- 4) = 768 - 24 {(- 4)}^{2} - 12 \cdot - 4$

Step 14.2.2.1.3

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

$f' (- 4) = 768 - 24 \cdot 16 - 12 \cdot - 4$

Step 14.2.2.1.4

Multiply $- 24$ by $16$ .

$f' (- 4) = 768 - 384 - 12 \cdot - 4$

Step 14.2.2.1.5

Multiply $- 12$ by $- 4$ .

$f' (- 4) = 768 - 384 + 48$

Step 14.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 14.2.2.2.1

Subtract $384$ from $768$ .

$f' (- 4) = 384 + 48$

Step 14.2.2.2.2

Add $384$ and $48$ .

$f' (- 4) = 432$

Step 14.2.2.3

The final answer is $432$ .

$432$

Step 14.3

Substitute any number, such as $- 0.5$ , from the interval $(- 1, 0)$ in the first derivative $- 12 x^{3} - 24 x^{2} - 12 x$ to check if the result is negative or positive.

Tap for more steps...

Step 14.3.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f' (- 0.5) = - 12 {(- 0.5)}^{3} - 24 {(- 0.5)}^{2} - 12 \cdot - 0.5$

Step 14.3.2

Simplify the result.

Tap for more steps...

Step 14.3.2.1

Simplify each term.

Tap for more steps...

Step 14.3.2.1.1

Raise $- 0.5$ to the power of $3$ .

$f' (- 0.5) = - 12 \cdot - 0.125 - 24 {(- 0.5)}^{2} - 12 \cdot - 0.5$

Step 14.3.2.1.2

Multiply $- 12$ by $- 0.125$ .

$f' (- 0.5) = 1.5 - 24 {(- 0.5)}^{2} - 12 \cdot - 0.5$

Step 14.3.2.1.3

Raise $- 0.5$ to the power of $2$ .

$f' (- 0.5) = 1.5 - 24 \cdot 0.25 - 12 \cdot - 0.5$

Step 14.3.2.1.4

Multiply $- 24$ by $0.25$ .

$f' (- 0.5) = 1.5 - 6 - 12 \cdot - 0.5$

Step 14.3.2.1.5

Multiply $- 12$ by $- 0.5$ .

$f' (- 0.5) = 1.5 - 6 + 6$

Step 14.3.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 14.3.2.2.1

Subtract $6$ from $1.5$ .

$f' (- 0.5) = - 4.5 + 6$

Step 14.3.2.2.2

Add $- 4.5$ and $6$ .

$f' (- 0.5) = 1.5$

Step 14.3.2.3

The final answer is $1.5$ .

$1.5$

Step 14.4

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $- 12 x^{3} - 24 x^{2} - 12 x$ to check if the result is negative or positive.

Tap for more steps...

Step 14.4.1

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

$f' (2) = - 12 {(2)}^{3} - 24 {(2)}^{2} - 12 \cdot 2$

Step 14.4.2

Simplify the result.

Tap for more steps...

Step 14.4.2.1

Simplify each term.

Tap for more steps...

Step 14.4.2.1.1

Raise $2$ to the power of $3$ .

$f' (2) = - 12 \cdot 8 - 24 {(2)}^{2} - 12 \cdot 2$

Step 14.4.2.1.2

Multiply $- 12$ by $8$ .

$f' (2) = - 96 - 24 {(2)}^{2} - 12 \cdot 2$

Step 14.4.2.1.3

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

$f' (2) = - 96 - 24 \cdot 4 - 12 \cdot 2$

Step 14.4.2.1.4

Multiply $- 24$ by $4$ .

$f' (2) = - 96 - 96 - 12 \cdot 2$

Step 14.4.2.1.5

Multiply $- 12$ by $2$ .

$f' (2) = - 96 - 96 - 24$

Step 14.4.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 14.4.2.2.1

Subtract $96$ from $- 96$ .

$f' (2) = - 192 - 24$

Step 14.4.2.2.2

Subtract $24$ from $- 192$ .

$f' (2) = - 216$

Step 14.4.2.3

The final answer is $- 216$ .

$- 216$

Step 14.5

Since the first derivative did not change signs around $x = - 1$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.6

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 15