Algebra Examples

$f (x) = 6 x^{4} - 2 x^{2} + 5 x - 1$ , $[0, 1]$

Step 1

Find the critical points.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $6 x^{4} - 2 x^{2} + 5 x - 1$ with respect to $x$ is $\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]$ .

$f' (x) = \frac{d}{d x} (6 x^{4}) + \frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (5 x) + \frac{d}{d x} (- 1)$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [6 x^{4}]$ .

Tap for more steps...

Step 1.1.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

$f' (x) = 6 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (5 x) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.2

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

$f' (x) = 6 (4 x^{3}) + \frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (5 x) + \frac{d}{d x} (- 1)$

Step 1.1.1.2.3

Multiply $4$ by $6$ .

$f' (x) = 24 x^{3} + \frac{d}{d x} (- 2 x^{2}) + \frac{d}{d x} (5 x) + \frac{d}{d x} (- 1)$

Step 1.1.1.3

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

Tap for more steps...

Step 1.1.1.3.1

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

$f' (x) = 24 x^{3} - 2 \frac{d}{d x} x^{2} + \frac{d}{d x} (5 x) + \frac{d}{d x} (- 1)$

Step 1.1.1.3.2

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

$f' (x) = 24 x^{3} - 2 (2 x) + \frac{d}{d x} (5 x) + \frac{d}{d x} (- 1)$

Step 1.1.1.3.3

Multiply $2$ by $- 2$ .

$f' (x) = 24 x^{3} - 4 x + \frac{d}{d x} (5 x) + \frac{d}{d x} (- 1)$

Step 1.1.1.4

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

Tap for more steps...

Step 1.1.1.4.1

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

$f' (x) = 24 x^{3} - 4 x + 5 \frac{d}{d x} (x) + \frac{d}{d x} (- 1)$

Step 1.1.1.4.2

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

$f' (x) = 24 x^{3} - 4 x + 5 \cdot 1 + \frac{d}{d x} (- 1)$

Step 1.1.1.4.3

Multiply $5$ by $1$ .

$f' (x) = 24 x^{3} - 4 x + 5 + \frac{d}{d x} (- 1)$

Step 1.1.1.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.1.5.1

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

$f' (x) = 24 x^{3} - 4 x + 5 + 0$

Step 1.1.1.5.2

Add $24 x^{3} - 4 x + 5$ and $0$ .

$f' (x) = 24 x^{3} - 4 x + 5$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $24 x^{3} - 4 x + 5$ .

$24 x^{3} - 4 x + 5$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $24 x^{3} - 4 x + 5 = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$24 x^{3} - 4 x + 5 = 0$

Step 1.2.2

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

$x \approx - 0.68586971$

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

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 1.4

Evaluate $6 x^{4} - 2 x^{2} + 5 x - 1$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at $x = - 0.68586971$ .

Tap for more steps...

Step 1.4.1.1

Substitute $- 0.68586971$ for $x$ .

$6 {(- 0.68586971)}^{4} - 2 {(- 0.68586971)}^{2} + 5 (- 0.68586971) - 1$

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Simplify each term.

Tap for more steps...

Step 1.4.1.2.1.1

Raise $- 0.68586971$ to the power of $4$ .

$6 \cdot 0.2212924 - 2 {(- 0.68586971)}^{2} + 5 (- 0.68586971) - 1$

Step 1.4.1.2.1.2

Multiply $6$ by $0.2212924$ .

$1.3277544 - 2 {(- 0.68586971)}^{2} + 5 (- 0.68586971) - 1$

Step 1.4.1.2.1.3

Raise $- 0.68586971$ to the power of $2$ .

$1.3277544 - 2 \cdot 0.47041726 + 5 (- 0.68586971) - 1$

Step 1.4.1.2.1.4

Multiply $- 2$ by $0.47041726$ .

$1.3277544 - 0.94083452 + 5 (- 0.68586971) - 1$

Step 1.4.1.2.1.5

Multiply $5$ by $- 0.68586971$ .

$1.3277544 - 0.94083452 - 3.42934856 - 1$

Step 1.4.1.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 1.4.1.2.2.1

Subtract $0.94083452$ from $1.3277544$ .

$0.38691987 - 3.42934856 - 1$

Step 1.4.1.2.2.2

Subtract $3.42934856$ from $0.38691987$ .

$- 3.04242868 - 1$

Step 1.4.1.2.2.3

Subtract $1$ from $- 3.04242868$ .

$- 4.04242868$

Step 1.4.2

List all of the points.

$(- 0.68586971, - 4.04242868)$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

Tap for more steps...

Step 3.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 3.1.1

Substitute $0$ for $x$ .

$6 {(0)}^{4} - 2 {(0)}^{2} + 5 (0) - 1$

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

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

$6 \cdot 0 - 2 {(0)}^{2} + 5 (0) - 1$

Step 3.1.2.1.2

Multiply $6$ by $0$ .

$0 - 2 {(0)}^{2} + 5 (0) - 1$

Step 3.1.2.1.3

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

$0 - 2 \cdot 0 + 5 (0) - 1$

Step 3.1.2.1.4

Multiply $- 2$ by $0$ .

$0 + 0 + 5 (0) - 1$

Step 3.1.2.1.5

Multiply $5$ by $0$ .

$0 + 0 + 0 - 1$

Step 3.1.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 3.1.2.2.1

Add $0$ and $0$ .

$0 + 0 - 1$

Step 3.1.2.2.2

Add $0$ and $0$ .

$0 - 1$

Step 3.1.2.2.3

Subtract $1$ from $0$ .

$- 1$

Step 3.2

Evaluate at $x = 1$ .

Tap for more steps...

Step 3.2.1

Substitute $1$ for $x$ .

$6 {(1)}^{4} - 2 {(1)}^{2} + 5 (1) - 1$

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

One to any power is one.

$6 \cdot 1 - 2 {(1)}^{2} + 5 (1) - 1$

Step 3.2.2.1.2

Multiply $6$ by $1$ .

$6 - 2 {(1)}^{2} + 5 (1) - 1$

Step 3.2.2.1.3

One to any power is one.

$6 - 2 \cdot 1 + 5 (1) - 1$

Step 3.2.2.1.4

Multiply $- 2$ by $1$ .

$6 - 2 + 5 (1) - 1$

Step 3.2.2.1.5

Multiply $5$ by $1$ .

$6 - 2 + 5 - 1$

Step 3.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 3.2.2.2.1

Subtract $2$ from $6$ .

$4 + 5 - 1$

Step 3.2.2.2.2

Add $4$ and $5$ .

$9 - 1$

Step 3.2.2.2.3

Subtract $1$ from $9$ .

$8$

Step 3.3

List all of the points.

$(0, - 1), (1, 8)$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(1, 8)$

Absolute Minimum: $(0, - 1)$

Step 5