Calculus Examples

$p (x) = 33 - \frac{x}{24}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $33 - \frac{x}{24}$ with respect to $x$ is $\frac{d}{d x} [33] + \frac{d}{d x} [- \frac{x}{24}]$ .

$\frac{d}{d x} [33] + \frac{d}{d x} [- \frac{x}{24}]$

Step 1.1.2

Since $33$ is constant with respect to $x$ , the derivative of $33$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{x}{24}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{x}{24}]$ .

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Step 1.2.1

Since $- \frac{1}{24}$ is constant with respect to $x$ , the derivative of $- \frac{x}{24}$ with respect to $x$ is $- \frac{1}{24} \frac{d}{d x} [x]$ .

$0 - \frac{1}{24} \frac{d}{d x} [x]$

Step 1.2.2

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

$0 - \frac{1}{24} \cdot 1$

Step 1.2.3

Multiply $- 1$ by $1$ .

$0 - \frac{1}{24}$

Step 1.3

Subtract $\frac{1}{24}$ from $0$ .

$- \frac{1}{24}$

Step 2

Since $- \frac{1}{24}$ is constant with respect to $x$ , the derivative of $- \frac{1}{24}$ with respect to $x$ is $0$ .

$p'' (x) = 0$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{1}{24} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6