Calculus Examples

$m (x) = \ln (x)$ , given $\frac{1}{10} \leq x \leq \frac{1}{5}$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

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

Step 1.1.2

The first derivative of $m (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{1}{x} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

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

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.

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Step 1.3.1

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.4

Evaluate $\ln (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$\ln (0)$

Step 1.4.1.2

The natural logarithm of zero is undefined.

Undefined

Step 1.5

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.

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Step 2.1

Substitute $\frac{1}{10}$ for $x$ .

$\ln (\frac{1}{10})$

Step 2.2

Substitute $\frac{1}{5}$ for $x$ .

$\ln (\frac{1}{5})$

Step 2.3

List all of the points.

$(\frac{1}{10}, \ln (\frac{1}{10})), (\frac{1}{5}, \ln (\frac{1}{5}))$

Step 3

Compare the $m (x)$ 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 $m (x)$ value and the minimum will occur at the lowest $m (x)$ value.

Absolute Maximum: $(\frac{1}{5}, \ln (\frac{1}{5}))$

Absolute Minimum: $(\frac{1}{10}, \ln (\frac{1}{10}))$

Step 4