Calculus Examples

$f (x) = x - (x)$ , $0 \leq x \leq 1$

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

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $x - (x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- (x)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- (x)]$

Step 1.1.1.1.2

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

$1 + \frac{d}{d x} [- (x)]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [- (x)]$ .

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Step 1.1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$1 - \frac{d}{d x} [x]$

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 $n = 1$ .

$1 - 1 \cdot 1$

Step 1.1.1.2.3

Multiply $- 1$ by $1$ .

$1 - 1$

Step 1.1.1.3

Subtract $1$ from $1$ .

$f' (x) = 0$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $0$ .

$0$

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

$0 = 0$

Step 1.2.2

Since $0 = 0$ , the equation will always be true.

Always true

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 $x = 0$ .

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Step 2.1.1

Substitute $0$ for $x$ .

$(0) - (0)$

Step 2.1.2

Subtract $0$ from $0$ .

$0$

Step 2.2

Evaluate at $x = 1$ .

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Step 2.2.1

Substitute $1$ for $x$ .

$(1) - (1)$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Multiply $- 1$ by $1$ .

$1 - 1$

Step 2.2.2.2

Subtract $1$ from $1$ .

$0$

Step 2.3

List all of the points.

$(0, 0), (1, 0)$

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: $(0, 0), (1, 0)$

Absolute Minimum: $(0, 0), (1, 0)$

Step 4