Question

$IfA={\sin }^{2}x+{\cos }^{4}x$, then for all real $x$:

• A. $\frac{3}{4}\le A\le 1$
• B. $\frac{13}{16}\le A\le 1$
• C. $1\le A\le 2$
• D. $\frac{3}{4}\le A\le \frac{13}{16}$

Answer 1

The correct option is A $\frac{3}{4}\le A\le 1$

$A={\sin }^{2}x+{\cos }^{4}x$

$={\sin }^{2}x+{\cos }^{2}x.(1-{\sin }^{2}x)$

$=1-\frac{1}{4}{\sin }^{2}2x$

Since; $0\le {\sin }^{2}2x\le 1$

So, $\frac{3}{4}\le A\le 1$

Hence, option 'A' is correct.