Mean Value Theorems — Maths STD 12 Science — Question

$\text{f}(\text{x})=\log\Big(\frac{\text{x}^2+2}{3}\Big)$

We know that, logarithmic function is differentiable and so continuous in its domain $\text{f}(\text{x})=\log\Big(\frac{\text{x}^2+2}{3}\Big)$ is continuous is $[-1,1]$ and differentiable is $(-1,1).$

Also,

f(1) = f(-1) = 0

Thus, f(x) satisfies all the conditions of Rolle's theorem.

Now, we have show that there must exists $\text{c}\in(-1,1)$ such that f'(c) = 0.

We have,

$\text{f}(\text{x})=\log\Big(\frac{\text{x}^2+2}{3}\Big)$

$\text{f}'(\text{x})=\frac{3(2\text{x})}{\text{x}^2+2}=\frac{6\text{x}}{\text{x}^2+2}$

$\therefore\ \text{f}'(\text{x})=0$

$\Rightarrow\frac{6\text{x}}{\text{x}^2+2}=0$

$\Rightarrow\text{x}=0$

Thus, $\text{c}=0\in(-1,1)$ such that f'(c) = 0.

Hence, Rolle's theorem is verified.