Calculus Examples

$y = 2 {(\frac{1}{5})}^{x}$ , $[- 2, 3]$

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

Since $2$ is constant with respect to $x$ , the derivative of $2 {(\frac{1}{5})}^{x}$ with respect to $x$ is $2 \frac{d}{d x} [{(\frac{1}{5})}^{x}]$ .

$2 \frac{d}{d x} [{(\frac{1}{5})}^{x}]$

Step 1.1.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $\frac{1}{5}$ .

$2 ({(\frac{1}{5})}^{x} \ln (\frac{1}{5}))$

Step 1.1.1.3

Simplify.

Tap for more steps...

Step 1.1.1.3.1

Apply the product rule to $\frac{1}{5}$ .

$2 \frac{1^{x}}{5^{x}} \ln (\frac{1}{5})$

Step 1.1.1.3.2

Combine terms.

Tap for more steps...

Step 1.1.1.3.2.1

One to any power is one.

$2 \frac{1}{5^{x}} \ln (\frac{1}{5})$

Step 1.1.1.3.2.2

Combine $2$ and $\frac{1}{5^{x}}$ .

$\frac{2}{5^{x}} \ln (\frac{1}{5})$

Step 1.1.1.3.2.3

Combine $\frac{2}{5^{x}}$ and $\ln (\frac{1}{5})$ .

$f' (x) = \frac{2 \ln (\frac{1}{5})}{5^{x}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 \ln (\frac{1}{5})}{5^{x}}$ .

$\frac{2 \ln (\frac{1}{5})}{5^{x}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{2 \ln (\frac{1}{5})}{5^{x}} = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{2 \ln (\frac{1}{5})}{5^{x}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$2 \ln (\frac{1}{5}) = 0$

Step 1.2.3

Solve the equation for $x$ .

Tap for more steps...

Step 1.2.3.1

Simplify $2 \log_{2.71828182} (\frac{1}{5})$ .

Tap for more steps...

Step 1.2.3.1.1

Simplify $2 \log_{2.71828182} (\frac{1}{5})$ by moving $2$ inside the logarithm.

$\log_{2.71828182} ({(\frac{1}{5})}^{2}) = 0$

Step 1.2.3.1.2

Apply the product rule to $\frac{1}{5}$ .

$\log_{2.71828182} (\frac{1^{2}}{5^{2}}) = 0$

Step 1.2.3.1.3

One to any power is one.

$\log_{2.71828182} (\frac{1}{5^{2}}) = 0$

Step 1.2.3.1.4

Raise $5$ to the power of $2$ .

$\log_{2.71828182} (\frac{1}{25}) = 0$

Step 1.2.3.2

Since $\log_{2.71828182} (\frac{1}{25}) \neq 0$ , 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

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

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 2

Evaluate at the included endpoints.

Tap for more steps...

Step 2.1

Evaluate at $x = - 2$ .

Tap for more steps...

Step 2.1.1

Substitute $- 2$ for $x$ .

$2 {(\frac{1}{5})}^{- 2}$

Step 2.1.2

Simplify.

Tap for more steps...

Step 2.1.2.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$2 \cdot 5^{2}$

Step 2.1.2.2

Raise $5$ to the power of $2$ .

$2 \cdot 25$

Step 2.1.2.3

Multiply $2$ by $25$ .

$50$

Step 2.2

Evaluate at $x = 3$ .

Tap for more steps...

Step 2.2.1

Substitute $3$ for $x$ .

$2 {(\frac{1}{5})}^{3}$

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

Simplify the expression.

Tap for more steps...

Step 2.2.2.1.1

Apply the product rule to $\frac{1}{5}$ .

$2 \frac{1^{3}}{5^{3}}$

Step 2.2.2.1.2

One to any power is one.

$2 \frac{1}{5^{3}}$

Step 2.2.2.1.3

Raise $5$ to the power of $3$ .

$2 (\frac{1}{125})$

Step 2.2.2.2

Combine $2$ and $\frac{1}{125}$ .

$\frac{2}{125}$

Step 2.3

List all of the points.

$(- 2, 50), (3, \frac{2}{125})$

Step 3

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: $(- 2, 50)$

Absolute Minimum: $(3, \frac{2}{125})$

Step 4