Calculus Examples

$f (x) = 3 x$ , $[1, 3]$

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

Find the first derivative.

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Step 1.1.1.1

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

$3 \frac{d}{d x} [x]$

Step 1.1.1.2

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

$3 \cdot 1$

Step 1.1.1.3

Multiply $3$ by $1$ .

$f' (x) = 3$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $3$ .

$3$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $3 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$3 = 0$

Step 1.2.2

Since $3 \neq 0$ , there are no solutions.

No solution

Step 1.3

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

Evaluate at $x = 1$ .

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Step 2.1.1

Substitute $1$ for $x$ .

$3 (1)$

Step 2.1.2

Multiply $3$ by $1$ .

$3$

Step 2.2

Evaluate at $x = 3$ .

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Step 2.2.1

Substitute $3$ for $x$ .

$3 (3)$

Step 2.2.2

Multiply $3$ by $3$ .

$9$

Step 2.3

List all of the points.

$(1, 3), (3, 9)$

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: $(3, 9)$

Absolute Minimum: $(1, 3)$

Step 4