Q. 45 Determine whether or not each fu... [FREE SOLUTION]

Chapter 3: Q. 45 (page 249)

Determine whether or not each function f in Exercises 41鈥�48 satisfies the hypotheses of Rolle鈥檚 Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c 鈭� (a, b) that satisfy the conclusion of Rolle鈥檚 Theorem.
$f (x) = \cos (x), [a, b] = [- \frac{蟺}{2}, \frac{3 蟺}{2}]$

Short Answer

Expert verified

The function $f (x) = \cos (x)$ satisfies the hypotheses of Rolle's theorem. Theexact values ofc that satisfies conclusion of Rolle's theorem are $c = 0, c = 蟺$ .

Step by step solution

Step 1. Given information.

Consider the given function $f (x) = \cos (x), [a, b] = [- \frac{蟺}{2}, \frac{3 蟺}{2}]$ .

Step 2. Satisfy hypotheses of Rolle's theorem.

Cosine function is continuous and differentiable everywhere in its domain. So, the given function is continuous on $[- \frac{蟺}{2}, \frac{3 蟺}{2}]$ and differentiable on $(- \frac{蟺}{2}, \frac{3 蟺}{2})$ .

Now, find $f (- \frac{蟺}{2})$ and $f (\frac{3 蟺}{2})$ .

$\begin{array}{rcl} f (- \frac{蟺}{2}) & = & \cos (- \frac{蟺}{2}) \\ = & 0 \\ f (\frac{3 蟺}{2}) & = & \cos (\frac{3 蟺}{2}) \\ = & 0 \end{array}$

So, $f (- \frac{蟺}{2}) = f (\frac{3 蟺}{2})$ .

Thus, the Rolle's theorem applies on $[- \frac{蟺}{2}, \frac{3 蟺}{2}]$ then there must exist some value $c 鈭� (- \frac{蟺}{2}, \frac{3 蟺}{2})$ such that $f' (c) = 0$ .

Step 3. Find the exact values of c.

The function is $f (c) = \cos (c)$ .

Differentiate the function with respect to c.

$f' (c) = - \sin (c)$

Solve $f' (c) = 0$ .

$\begin{array}{rcl} - \sin (c) & = & 0 \\ \sin (c) & = & 0 \\ c & = & 0, 蟺 \end{array}$

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Satisfy hypotheses of Rolle's theorem.

Step 3. Find the exact values of c.

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