Functions Question 591
Question: If $ f(x)={{\sin }^{2}}x+{{\sin }^{2}}( x+\frac{\pi }{3} )+\cos x\cos ( x+\frac{\pi }{3} ) $ and $ g( \frac{5}{4} )=1 $ , then $ (gof)(x)= $
[IIT 1996]
Options:
A) -2
B) -1
C) 2
D) 1
Show Answer
Answer:
Correct Answer: D
Solution:
$ {f}’(x)=2,\sin x\cos x+2,\sin ,( x+\frac{\pi }{3} ),\cos ,( x+\frac{\pi }{3} ) $ $ -\sin x\cos ,( x+\frac{\pi }{3} )-\cos x\sin ( x+\frac{\pi }{3} ) $
$ =\sin 2x+\sin ,( 2x+\frac{2\pi }{3} )-\sin ,( x+x+\frac{\pi }{3} ) $
$ =2\sin ,( 2x+\frac{2\pi }{3} ),\cos ,( \frac{\pi }{3} )-\sin ,( 2x+\frac{\pi }{3} )=0 $
$ \Rightarrow f(x)=k, $ where k is a constant.
But $ f(0)={{\sin }^{2}}0+{{\sin }^{2}}( \frac{\pi }{3} )+\cos 0\cos ,( \frac{\pi }{3} )=\frac{5}{4} $
Thus $ f(x)=\frac{5}{4},,\forall ,x\in R. $
Therefore, $ (gof)(x)=g,[f(x),]=g,( \frac{5}{4} )=1. $
BETA
