Relations Question 1
Question: The function $ f:R\to R $ is defined by $ f( x )={{\cos }^{2}}x+{{\sin }^{4}}x $ for $ x\in R $ . Then the range of $ f(x) $ is
Options:
A) $ ( \frac{3}{4},1 ] $
B) $ [ \frac{3}{4},1 ) $
C) $ [ \frac{3}{4},1 ] $
D) $ ( \frac{3}{4},1 ) $
Show Answer
Answer:
Correct Answer: C
Solution:
$ y=f(x)=cos^{2}x+{{\sin }^{4}}x $
$ ={{\cos }^{2}}x+{{\sin }^{2}}x(1-cos^{2}x) $
$ ={{\cos }^{2}}x+{{\sin }^{2}}x-{{\sin }^{2}}xcos^{2}x) $
$ =1-{{\sin }^{2}}x{{\cos }^{2}}x $
$ =1-\frac{1}{4}{{\sin }^{2}}2x $
$ \therefore \frac{3}{4}\le f(x)\le 1 $
$ (\therefore 0\le sin^{2}2x\le 1) $
$ \therefore f(x)\in [3/4,1] $
BETA
