Q. 48 - Calculus

Question

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

$f\left(x\right)=\ln \left|x^2-1\right|, \left[a, b\right]=\left[-\sqrt{2}, \sqrt{2}\right]$

Step-by-Step Solution

Answer

The function $f\left(x\right)=\ln \left|x^2-1\right|$does not satisfies the hypotheses of Rolle's theorem.

Step 1. Given information.

Consider the given function $f\left(x\right)=\ln \left|x^2-1\right|, \left[a, b\right]=\left[-\sqrt{2}, \sqrt{2}\right]$.

Step 2. Satisfy hypotheses of Rolle's theorem.

The function $f\left(x\right)=\ln \left|x^2-1\right|$ is not defined at $x=-1, 1$.

So, the function is continuous at $x=-1, 1$ in the interval $\left[0, 2\right]$.

Thus, the given function des not satisfy the hypotheses of Rolle's theorem.