If $A=\sin ^2 x+\cos ^4 x$, then for all real $x$

If $A=\sin ^2 x+\cos ^4 x$, then for all real $x$
  1. $\frac{13}{16} \leq \mathrm{A} \leq 1$
  2. $1 \leq \mathrm{A} \leq 2$
  3. $\frac{3}{4} \leq \mathrm{A} \leq \frac{13}{16}$
  4. $\frac{3}{4} \leq \mathrm{A} \leq 1$

Solution

$A=\sin ^2 x+\cos ^4 x=\frac{7+\cos 4 x}{8} \Rightarrow \frac{3}{4} \leq A \leq 1$

Asked in: JEE Main 2011

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