Question

Choose the correct answer from the alternative given.
The maximum value of ${\sin }^4\theta +{\cos }^4\theta$ is:

Answer 1

Given

${\sin }^4\theta +{\cos }^4\theta$

$({\sin }^2\theta {)}^2+({\cos }^2\theta {)}^2$

$({\sin }^2\theta +{\cos }^2\theta {)}^2-2{\sin }^2\theta {\cos }^2\theta$

$1-2{\sin }^2\theta {\cos }^2\theta$

The expression has max value at $x$ if $f^{\prime }\left(x\right)=0$

$⟹\frac{d}{d\theta }(1-2{\sin }^2\theta {\cos }^2\theta )=0$

$⟹\frac{1}{2}\frac{d}{d\theta }\left({\sin }^{2}2\theta \right)=0$

$⟹\frac{1}{2}\times 2\sin 2\theta \left(2\cos 2\theta \right)=0$

$⟹2\sin 2\theta \cos 2\theta =0$

$⟹\sin 4\theta =0$

$⟹\theta =0$

Max value is ${\sin }^4\left(0\right)+{\cos }^4\left(0\right)$

$0+1=1$