જો $f:[-5,5] \rightarrow \mathrm{R}$ વિકલનીય વિધેય હોય અને $f^{\prime}(x)$ ક્યાંય શૂન્ય ના બને તો સાબિત કરો કે $f(-5) \neq f(5).$

It is given that $f:[-5,5] \rightarrow R$ is a differentiable function.

Since every differentiable function is a continuous function, we obtain

a) $f$ is continuous on $[-5,5].$

b) $f$ is continuous on $(-5,5).$

Therefore, by the Mean Value Theorem, there exists $c \in(-5,5)$ such that

$f^{\prime}(c)=\frac{f(5)-f(-5)}{5-(-5)}$

$\Rightarrow 10 f^{\prime}(c)=f(5)-f(-5)$

It is also given that $f^{\prime}(x)$ does not vanish anywhere.

$\therefore f^{\prime}(c) \neq 0$

$\Rightarrow 10 f^{\prime}(c) \neq 0$

$\Rightarrow f(5)-f(-5) \neq 0$

$\Rightarrow f(5) \neq f(-5)$

Hence, proved.