#### Question

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin^{4} x + cos^{4} x on \[\left[ 0, \frac{\pi}{2} \right]\] ?

#### Solution

The given function is \[f\left( x \right) = \sin^4 x + \cos^4 x\] .

Since

\[\sin x \text { and } \cos x\] are everywhere continuous and differentiable,

**.**

\[f\left( x \right) = \sin^4 x + \cos^4 x\]

\[ \Rightarrow f'\left( x \right) = 4 \sin^3 x\cos x - 4 \cos^3 x\sin x\]

\[\therefore f'\left( x \right) = 0\]

\[ \Rightarrow 4 \sin^3 x\cos x - 4 \cos^3 x\sin x = 0\]

\[ \Rightarrow \sin^3 x\cos x - \cos^3 x\sin x = 0\]

\[ \Rightarrow \tan^3 x - \tan x = 0\]

\[ \Rightarrow \tan x\left( \tan^2 x - 1 \right) = 0\]

\[ \Rightarrow \tan x = 0, \tan^2 x = 1\]

\[ \Rightarrow \tan x = 0, \tan x = \pm 1\]

\[ \Rightarrow x = 0, x = \frac{\pi}{4}, \frac{3\pi}{4}\]

Thus

\[c = \frac{\pi}{4} \in \left( 0, \frac{\pi}{2} \right)\] such that \[f'\left( c \right) = 0\] .