Application Of Derivatives Ques 30
30. 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) $\frac{5 \pi}{8}<x<\frac{3 \pi}{4}$
$(1999,2 \mathrm{M})$
Show Answer
Answer:
(c)
Solution:
Formula:
Increasing and decreasing of a function:
- 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 2x \cos 2x \\ & =-\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} ……(i) \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), it is correct.
BETA
