Application of Derivatives 2 Question 13
13. The function $f(x)=\sin ^{4} x+\cos ^{4} x$ increases, if
(a) $0<x<\frac{\pi}{8}$
(b) $\frac{\pi}{4}<x<\frac{3 \pi}{8}$
(c) $\frac{3 \pi}{8}<x<\frac{5 \pi}{8}$
(d) $\frac{5 \pi}{8}<x<\frac{3 \pi}{4}$
$(1999,2 M)$
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Solution:
- Given, $f(x)=\sin ^{4} x+\cos ^{4} x$
On differentiating w.r.t. $x$, we get
$$ \begin{aligned} f^{\prime}(x) & =4 \sin ^{3} x \cos x-4 \cos ^{3} x \sin x \\ & =4 \sin x \cos x\left(\sin ^{2} x-\cos ^{2} x\right) \\ & =2 \sin 2 x(-\cos 2 x) \\ & =-\sin 4 x \end{aligned} $$
Now, $\quad f^{\prime}(x)>0$, if $\sin 4 x<0$
$$ \begin{array}{ll} \Rightarrow & \pi<4 x<2 \pi \\ \Rightarrow & \frac{\pi}{4}<x<\frac{\pi}{2} \end{array} $$
$\Rightarrow$ Option (a) is not proper subset of Eq. (i), so it is not correct.
Now,
$$ \frac{\pi}{4}<x<\frac{3 \pi}{8} $$
Since, option (b) is the proper subset of Eq. (i), so it is correct.
