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Q. 59

Expert-verified
Found in: Page 249

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $\left[a,b\right]$. For those that do, use derivatives and algebra to find the exact values of all $c\in \left(a,b\right)$ that satisfy the conclusion of the Mean Value Theorem.$f\left(x\right)=\mathrm{sin}\left(x\right),\left[a,b\right]=\left[0,\frac{\mathrm{\pi }}{2}\right]$.

The function $f\left(x\right)=\mathrm{sin}\left(x\right)$ satisfies the Mean Value Theorem and the value is, $c={\mathrm{cos}}^{-1}\left(\frac{2}{\mathrm{\pi }}\right)$.

See the step by step solution

## Step 1. Given Information.

The given function is,

$f\left(x\right)=\mathrm{sin}\left(x\right),\left[a,b\right]=\left[0,\frac{\mathrm{\pi }}{2}\right]$.

## Step 2. Proving Mean Value Theorem.

The function $f\left(x\right)=\mathrm{sin}\left(x\right)$ is continuous and differentiable on $\left[0,\frac{\mathrm{\pi }}{2}\right]$. The Mean Value Theorem applies to this function on the interval $\left[0,\frac{\mathrm{\pi }}{2}\right]$.

The slope of the line from $\left(0,f\left(0\right)\right)$ to $\left(\frac{\mathrm{\pi }}{2},f\left(\frac{\mathrm{\pi }}{2}\right)\right)$ is:

By the Mean Value Theorem, there must exist at least one point $c\in \left(0,\frac{\mathrm{\pi }}{2}\right)$ with ${f}^{\text{'}}\left(c\right)=\frac{2}{\mathrm{\pi }}$

We have to find the value of $c$ with ${f}^{\text{'}}\left(c\right)=\frac{2}{\mathrm{\pi }}$ we solve it:

$\mathrm{cos}\left(c\right)=\frac{2}{\mathrm{\pi }}\phantom{\rule{0ex}{0ex}}⇒c={\mathrm{cos}}^{-1}\left(\frac{2}{\mathrm{\pi }}\right)$