A) \[\left( \frac{3}{4},\ 1 \right]\]
B) \[\left[ \frac{3}{4},\ 1 \right)\]
C) \[\left[ \frac{3}{4},\ 1 \right]\]
D) \[\left( \frac{3}{4},\ 1 \right)\]
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 \[f(R)\in [3/4,\,\,1]\].
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