Calculus Examples

$f (x) = x^{5} - 4 x^{3} + x^{2} + 2$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $x^{5} - 4 x^{3} + x^{2} + 2$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$5 x^{4} - 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 1.2.2

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

$5 x^{4} - 4 (3 x^{2}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 1.2.3

Multiply $3$ by $- 4$ .

$5 x^{4} - 12 x^{2} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 1.3

Differentiate.

Tap for more steps...

Step 1.3.1

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

$5 x^{4} - 12 x^{2} + 2 x + \frac{d}{d x} [2]$

Step 1.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$5 x^{4} - 12 x^{2} + 2 x + 0$

Step 1.3.3

Add $5 x^{4} - 12 x^{2} + 2 x$ and $0$ .

$5 x^{4} - 12 x^{2} + 2 x$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $5 x^{4} - 12 x^{2} + 2 x$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [2 x]$ .

$f'' (x) = \frac{d}{d x} (5 x^{4}) + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (2 x)$

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

Tap for more steps...

Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = 5 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (2 x)$

Step 2.2.2

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

$f'' (x) = 5 (4 x^{3}) + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (2 x)$

Step 2.2.3

Multiply $4$ by $5$ .

$f'' (x) = 20 x^{3} + \frac{d}{d x} (- 12 x^{2}) + \frac{d}{d x} (2 x)$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$f'' (x) = 20 x^{3} - 12 \frac{d}{d x} x^{2} + \frac{d}{d x} (2 x)$

Step 2.3.2

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

$f'' (x) = 20 x^{3} - 12 (2 x) + \frac{d}{d x} (2 x)$

Step 2.3.3

Multiply $2$ by $- 12$ .

$f'' (x) = 20 x^{3} - 24 x + \frac{d}{d x} (2 x)$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$f'' (x) = 20 x^{3} - 24 x + 2 \frac{d}{d x} (x)$

Step 2.4.2

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

$f'' (x) = 20 x^{3} - 24 x + 2 \cdot 1$

Step 2.4.3

Multiply $2$ by $1$ .

$f'' (x) = 20 x^{3} - 24 x + 2$

Step 3

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

$5 x^{4} - 12 x^{2} + 2 x = 0$

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

Differentiate.

Tap for more steps...

Step 4.1.1.1

By the Sum Rule, the derivative of $x^{5} - 4 x^{3} + x^{2} + 2$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 4.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

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

$5 x^{4} - 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

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 $n = 3$ .

$5 x^{4} - 4 (3 x^{2}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 4.1.2.3

Multiply $3$ by $- 4$ .

$5 x^{4} - 12 x^{2} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 4.1.3

Differentiate.

Tap for more steps...

Step 4.1.3.1

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

$5 x^{4} - 12 x^{2} + 2 x + \frac{d}{d x} [2]$

Step 4.1.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$5 x^{4} - 12 x^{2} + 2 x + 0$

Step 4.1.3.3

Add $5 x^{4} - 12 x^{2} + 2 x$ and $0$ .

$f' (x) = 5 x^{4} - 12 x^{2} + 2 x$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $5 x^{4} - 12 x^{2} + 2 x$ .

$5 x^{4} - 12 x^{2} + 2 x$

Step 5

Set the first derivative equal to $0$ then solve the equation $5 x^{4} - 12 x^{2} + 2 x = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$5 x^{4} - 12 x^{2} + 2 x = 0$

Step 5.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.62662472, 0, 0.16866593, 1.45795878$

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.

$x = - 1.62662472, 0, 0.16866593, 1.45795878$

Step 8

Evaluate the second derivative at $x = - 1.62662472$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$20 {(- 1.62662472)}^{3} - 24 \cdot - 1.62662472 + 2$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Raise $- 1.62662472$ to the power of $3$ .

$20 \cdot - 4.30389932 - 24 \cdot - 1.62662472 + 2$

Step 9.1.2

Multiply $20$ by $- 4.30389932$ .

$- 86.07798657 - 24 \cdot - 1.62662472 + 2$

Step 9.1.3

Multiply $- 24$ by $- 1.62662472$ .

$- 86.07798657 + 39.03899328 + 2$

Step 9.2

Simplify by adding numbers.

Tap for more steps...

Step 9.2.1

Add $- 86.07798657$ and $39.03899328$ .

$- 47.03899328 + 2$

Step 9.2.2

Add $- 47.03899328$ and $2$ .

$- 45.03899328$

Step 10

$x = - 1.62662472$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 1.62662472$ is a local maximum

Step 11

Find the y-value when $x = - 1.62662472$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $- 1.62662472$ in the expression.

$f (- 1.62662472) = {(- 1.62662472)}^{5} - 4 {(- 1.62662472)}^{3} + {(- 1.62662472)}^{2} + 2$

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

Raise $- 1.62662472$ to the power of $5$ .

$f (- 1.62662472) = - 11.38772158 - 4 {(- 1.62662472)}^{3} + {(- 1.62662472)}^{2} + 2$

Step 11.2.1.2

Raise $- 1.62662472$ to the power of $3$ .

$f (- 1.62662472) = - 11.38772158 - 4 \cdot - 4.30389932 + {(- 1.62662472)}^{2} + 2$

Step 11.2.1.3

Multiply $- 4$ by $- 4.30389932$ .

$f (- 1.62662472) = - 11.38772158 + 17.21559731 + {(- 1.62662472)}^{2} + 2$

Step 11.2.1.4

Raise $- 1.62662472$ to the power of $2$ .

$f (- 1.62662472) = - 11.38772158 + 17.21559731 + 2.64590798 + 2$

Step 11.2.2

Simplify by adding numbers.

Tap for more steps...

Step 11.2.2.1

Add $- 11.38772158$ and $17.21559731$ .

$f (- 1.62662472) = 5.82787573 + 2.64590798 + 2$

Step 11.2.2.2

Add $5.82787573$ and $2.64590798$ .

$f (- 1.62662472) = 8.47378371 + 2$

Step 11.2.2.3

Add $8.47378371$ and $2$ .

$f (- 1.62662472) = 10.47378371$

Step 11.2.3

The final answer is $10.47378371$ .

$y = 10.47378371$

Step 12

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.

$20 {(0)}^{3} - 24 \cdot 0 + 2$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

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

$20 \cdot 0 - 24 \cdot 0 + 2$

Step 13.1.2

Multiply $20$ by $0$ .

$0 - 24 \cdot 0 + 2$

Step 13.1.3

Multiply $- 24$ by $0$ .

$0 + 0 + 2$

Step 13.2

Simplify by adding numbers.

Tap for more steps...

Step 13.2.1

Add $0$ and $0$ .

$0 + 2$

Step 13.2.2

Add $0$ and $2$ .

$2$

Step 14

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

$x = 0$ is a local minimum

Step 15

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

Tap for more steps...

Step 15.1

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

$f (0) = {(0)}^{5} - 4 {(0)}^{3} + {(0)}^{2} + 2$

Step 15.2

Simplify the result.

Tap for more steps...

Step 15.2.1

Simplify each term.

Tap for more steps...

Step 15.2.1.1

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

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

Step 15.2.1.2

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

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

Step 15.2.1.3

Multiply $- 4$ by $0$ .

$f (0) = 0 + 0 + {(0)}^{2} + 2$

Step 15.2.1.4

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

$f (0) = 0 + 0 + 0 + 2$

Step 15.2.2

Simplify by adding numbers.

Tap for more steps...

Step 15.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 2$

Step 15.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 2$

Step 15.2.2.3

Add $0$ and $2$ .

$f (0) = 2$

Step 15.2.3

The final answer is $2$ .

$y = 2$

Step 16

Evaluate the second derivative at $x = 0.16866593$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$20 {(0.16866593)}^{3} - 24 \cdot 0.16866593 + 2$

Step 17

Evaluate the second derivative.

Tap for more steps...

Step 17.1

Simplify each term.

Tap for more steps...

Step 17.1.1

Raise $0.16866593$ to the power of $3$ .

$20 \cdot 0.00479824 - 24 \cdot 0.16866593 + 2$

Step 17.1.2

Multiply $20$ by $0.00479824$ .

$0.09596483 - 24 \cdot 0.16866593 + 2$

Step 17.1.3

Multiply $- 24$ by $0.16866593$ .

$0.09596483 - 4.0479824 + 2$

Step 17.2

Simplify by adding and subtracting.

Tap for more steps...

Step 17.2.1

Subtract $4.0479824$ from $0.09596483$ .

$- 3.95201757 + 2$

Step 17.2.2

Add $- 3.95201757$ and $2$ .

$- 1.95201757$

Step 18

$x = 0.16866593$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0.16866593$ is a local maximum

Step 19

Find the y-value when $x = 0.16866593$ .

Tap for more steps...

Step 19.1

Replace the variable $x$ with $0.16866593$ in the expression.

$f (0.16866593) = {(0.16866593)}^{5} - 4 {(0.16866593)}^{3} + {(0.16866593)}^{2} + 2$

Step 19.2

Simplify the result.

Tap for more steps...

Step 19.2.1

Simplify each term.

Tap for more steps...

Step 19.2.1.1

Raise $0.16866593$ to the power of $5$ .

$f (0.16866593) = 0.0001365 - 4 {(0.16866593)}^{3} + {(0.16866593)}^{2} + 2$

Step 19.2.1.2

Raise $0.16866593$ to the power of $3$ .

$f (0.16866593) = 0.0001365 - 4 \cdot 0.00479824 + {(0.16866593)}^{2} + 2$

Step 19.2.1.3

Multiply $- 4$ by $0.00479824$ .

$f (0.16866593) = 0.0001365 - 0.01919296 + {(0.16866593)}^{2} + 2$

Step 19.2.1.4

Raise $0.16866593$ to the power of $2$ .

$f (0.16866593) = 0.0001365 - 0.01919296 + 0.02844819 + 2$

Step 19.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 19.2.2.1

Subtract $0.01919296$ from $0.0001365$ .

$f (0.16866593) = - 0.01905646 + 0.02844819 + 2$

Step 19.2.2.2

Add $- 0.01905646$ and $0.02844819$ .

$f (0.16866593) = 0.00939173 + 2$

Step 19.2.2.3

Add $0.00939173$ and $2$ .

$f (0.16866593) = 2.00939173$

Step 19.2.3

The final answer is $2.00939173$ .

$y = 2.00939173$

Step 20

Evaluate the second derivative at $x = 1.45795878$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$20 {(1.45795878)}^{3} - 24 \cdot 1.45795878 + 2$

Step 21

Evaluate the second derivative.

Tap for more steps...

Step 21.1

Simplify each term.

Tap for more steps...

Step 21.1.1

Raise $1.45795878$ to the power of $3$ .

$20 \cdot 3.09910108 - 24 \cdot 1.45795878 + 2$

Step 21.1.2

Multiply $20$ by $3.09910108$ .

$61.98202173 - 24 \cdot 1.45795878 + 2$

Step 21.1.3

Multiply $- 24$ by $1.45795878$ .

$61.98202173 - 34.99101086 + 2$

Step 21.2

Simplify by adding and subtracting.

Tap for more steps...

Step 21.2.1

Subtract $34.99101086$ from $61.98202173$ .

$26.99101087 + 2$

Step 21.2.2

Add $26.99101087$ and $2$ .

$28.99101087$

Step 22

$x = 1.45795878$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 1.45795878$ is a local minimum

Step 23

Find the y-value when $x = 1.45795878$ .

Tap for more steps...

Step 23.1

Replace the variable $x$ with $1.45795878$ in the expression.

$f (1.45795878) = {(1.45795878)}^{5} - 4 {(1.45795878)}^{3} + {(1.45795878)}^{2} + 2$

Step 23.2

Simplify the result.

Tap for more steps...

Step 23.2.1

Simplify each term.

Tap for more steps...

Step 23.2.1.1

Raise $1.45795878$ to the power of $5$ .

$f (1.45795878) = 6.58758507 - 4 {(1.45795878)}^{3} + {(1.45795878)}^{2} + 2$

Step 23.2.1.2

Raise $1.45795878$ to the power of $3$ .

$f (1.45795878) = 6.58758507 - 4 \cdot 3.09910108 + {(1.45795878)}^{2} + 2$

Step 23.2.1.3

Multiply $- 4$ by $3.09910108$ .

$f (1.45795878) = 6.58758507 - 12.39640434 + {(1.45795878)}^{2} + 2$

Step 23.2.1.4

Raise $1.45795878$ to the power of $2$ .

$f (1.45795878) = 6.58758507 - 12.39640434 + 2.12564382 + 2$

Step 23.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 23.2.2.1

Subtract $12.39640434$ from $6.58758507$ .

$f (1.45795878) = - 5.80881926 + 2.12564382 + 2$

Step 23.2.2.2

Add $- 5.80881926$ and $2.12564382$ .

$f (1.45795878) = - 3.68317544 + 2$

Step 23.2.2.3

Add $- 3.68317544$ and $2$ .

$f (1.45795878) = - 1.68317544$

Step 23.2.3

The final answer is $- 1.68317544$ .

$y = - 1.68317544$

Step 24

These are the local extrema for $f (x) = x^{5} - 4 x^{3} + x^{2} + 2$ .

$(- 1.62662472, 10.47378371)$ is a local maxima

$(0, 2)$ is a local minima

$(0.16866593, 2.00939173)$ is a local maxima

$(1.45795878, - 1.68317544)$ is a local minima

Step 25