Calculus Examples

f (x) = 2 \sqrt{x} + 1

$f (x) = 2 \sqrt{x} + 1$ ;

$1 \leq x \leq 4$

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

$2 \sqrt{x} + 1$ with respect to

$x$ is

$\frac{d}{d x} [2 \sqrt{x}] + \frac{d}{d x} [1]$ .

$f' (x) = \frac{d}{d x} (2 \sqrt{x}) + \frac{d}{d x} (1)$

Step 1.1.1.2

Evaluate

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

Tap for more steps...

Step 1.1.1.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

$x^{\frac{1}{2}}$ .

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

Step 1.1.1.2.2

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x^{\frac{1}{2}}$ with respect to

$x$ is

$2 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 1.1.1.2.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = \frac{1}{2}$ .

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

Step 1.1.1.2.4

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 1.1.1.2.5

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 1.1.1.2.6

Combine the numerators over the common denominator.

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

Step 1.1.1.2.7

Simplify the numerator.

Tap for more steps...

Step 1.1.1.2.7.1

Multiply

$- 1$ by

$2$ .

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

Step 1.1.1.2.7.2

Subtract

$2$ from

$1$ .

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

Step 1.1.1.2.8

Move the negative in front of the fraction.

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

Step 1.1.1.2.9

Combine

$\frac{1}{2}$ and

$x^{- \frac{1}{2}}$ .

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

Step 1.1.1.2.10

Combine

$2$ and

$\frac{x^{- \frac{1}{2}}}{2}$ .

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

Step 1.1.1.2.11

Move

$x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.1.2.12

Cancel the common factor.

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

Step 1.1.1.2.13

Rewrite the expression.

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

Step 1.1.1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.1.3.1

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$f' (x) = \frac{1}{x^{\frac{1}{2}}} + 0$

Step 1.1.1.3.2

Add

$\frac{1}{x^{\frac{1}{2}}}$ and

$0$ .

$f' (x) = \frac{1}{x^{\frac{1}{2}}}$

Step 1.1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$\frac{1}{x^{\frac{1}{2}}}$ .

$\frac{1}{x^{\frac{1}{2}}}$

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

$\frac{1}{x^{\frac{1}{2}}} = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to

$0$ .

$\frac{1}{x^{\frac{1}{2}}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$1 = 0$

Step 1.2.3

Since

$1 \neq 0$ , there are no solutions.

No solution

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 1.3.1.1

Apply the rule

$x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{\sqrt{x^{1}}}$

Step 1.3.1.2

Anything raised to

$1$ is the base itself.

$\frac{1}{\sqrt{x}}$

Step 1.3.2

Set the denominator in

$\frac{1}{\sqrt{x}}$ equal to

$0$ to find where the expression is undefined.

$\sqrt{x} = 0$

Step 1.3.3

Solve for

$x$ .

Tap for more steps...

Step 1.3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = 0^{2}$

Step 1.3.3.2

Simplify each side of the equation.

Tap for more steps...

Step 1.3.3.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

$x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2

Simplify the left side.

Tap for more steps...

Step 1.3.3.2.2.1

Simplify

${(x^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.3.3.2.2.1.1

Multiply the exponents in

${(x^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.3.3.2.2.1.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.1.2

Cancel the common factor of

$2$ .

Tap for more steps...

Step 1.3.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 1.3.3.2.2.1.2

Simplify.

$x = 0^{2}$

Step 1.3.3.2.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$x = 0$

Step 1.3.4

Set the radicand in

$\sqrt{x}$ less than

$0$ to find where the expression is undefined.

$x < 0$

Step 1.3.5

The equation is undefined where the denominator equals

$0$ , the argument of a square root is less than

$0$ , or the argument of a logarithm is less than or equal to

$0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 1.4

Evaluate

$2 \sqrt{x} + 1$ at each

$x$ value where the derivative is

$0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at

$x = 0$ .

Tap for more steps...

Step 1.4.1.1

Substitute

$0$ for

$x$ .

$2 \sqrt{0} + 1$

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Remove parentheses.

$2 \sqrt{0} + 1$

Step 1.4.1.2.2

Simplify each term.

Tap for more steps...

Step 1.4.1.2.2.1

Rewrite

$0$ as

$0^{2}$ .

$2 \sqrt{0^{2}} + 1$

Step 1.4.1.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$2 \cdot 0 + 1$

Step 1.4.1.2.2.3

Multiply

$2$ by

$0$ .

$0 + 1$

Step 1.4.1.2.3

Add

$0$ and

$1$ .

$1$

Step 1.4.2

List all of the points.

$(0, 1)$

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

$x = 1$ .

Tap for more steps...

Step 3.1.1

Substitute

$1$ for

$x$ .

$2 \sqrt{1} + 1$

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

Remove parentheses.

$2 \sqrt{1} + 1$

Step 3.1.2.2

Simplify each term.

Tap for more steps...

Step 3.1.2.2.1

Any root of

$1$ is

$1$ .

$2 \cdot 1 + 1$

Step 3.1.2.2.2

Multiply

$2$ by

$1$ .

$2 + 1$

Step 3.1.2.3

Add

$2$ and

$1$ .

$3$

Step 3.2

Evaluate at

$x = 4$ .

Tap for more steps...

Step 3.2.1

Substitute

$4$ for

$x$ .

$2 \sqrt{4} + 1$

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

Remove parentheses.

$2 \sqrt{4} + 1$

Step 3.2.2.2

Simplify each term.

Tap for more steps...

Step 3.2.2.2.1

Rewrite

$4$ as

$2^{2}$ .

$2 \sqrt{2^{2}} + 1$

Step 3.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$2 \cdot 2 + 1$

Step 3.2.2.2.3

Multiply

$2$ by

$2$ .

$4 + 1$

Step 3.2.2.3

Add

$4$ and

$1$ .

$5$

Step 3.3

List all of the points.

$(1, 3), (4, 5)$

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:

$(4, 5)$

Absolute Minimum:

$(1, 3)$

Step 5