Verify the Rolle’s theorem for each of the functions: f(x)= (x^2+… - Vidyadip

Question

Verify the Rolle’s theorem for each of the functions:
$\text{f(x)}=\log(\text{x}^2+2)-\log3\text{ in }[-1,1].$

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Answer

We have, $\text{f(x)}=\log(\text{x}^2+2)-\log3$

Logarithmic functions are continuous in their domain.

Hence, $\text{f(x)}=\log(\text{x}^2+2)-\log3$ is continuous in $[-1,1]$

$\text{f}'(\text{x})=\frac{1}{\text{x}^2+2}\cdot2\text{x}-0$

$=\frac{2\text{x}}{\text{x}^2+2},$ which exists in $(-1,1).$

Hence, f(x) is differentiable in $(-1,1).$

$\text{f}(-1)=\log\big[(-1)^2+2\big]-\log3=\log3-\log3=0$ and

$\text{f}(1)=\log(1^2+2)-\log3=\log3-\log3=0$

$\Rightarrow\ \text{f}(-1)=\text{f}(1)$

Conditions of Rolle’s theorem are satisfied.

Hence, there exists a real number c such that

$\text{f}'(\text{c})=0$

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

$\Rightarrow\ \text{c}=0\in(-1,1)$

Hence, Rolle’s theorem has been verified.

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