Functions Question 657
Question: The function $ f:R\to R $ is defined by $ f(x)={{\cos }^{2}}x+{{\sin }^{4}}x $ for $ x\in R $ , then $ f(R)= $
[EAMCET 2002]
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 $
Þ $ y=f(x)={{\cos }^{2}}x+{{\sin }^{2}}x(1-{{\cos }^{2}}x) $
Þ $ y={{\cos }^{2}}x+{{\sin }^{2}}x-{{\sin }^{2}}x{{\cos }^{2}}x $
Þ $ y=1-{{\sin }^{2}}x{{\cos }^{2}}x $
Þ $ y=1-\frac{1}{4}.{{\sin }^{2}}2x $ \ $ \frac{3}{4}\le f(x)\le 1 $ , $ (\because 0\le {{\sin }^{2}}2x\le 1) $
Þ $ f(R)\in [3/4,1] $ .
BETA
