If A = ^2 x + ^4 x,then for all real x : - Vidyadip

MCQ

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

A
$1 \le A \le 2$
B
$\frac{{13}}{{16}} \le A \le 1$
✓
$\frac{3}{4} \le A \le 1$
D
$\frac{3}{4} \le A \le \frac{{13}}{{16}}$

✓

Answer

Correct option: C.

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

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

We have $\cos ^{4} x \leq \cos ^{2} x$

$\sin ^{2} x=\sin ^{2} x$

Adding $\sin ^{2} x+\cos ^{4} x \leq \sin ^{2} x+\cos ^{2} x$

$\therefore A \leq 1$

Again $A=t+(1-t)^{2}=t^{2}-t+1, t \geq 0$,

where minimum is $3 / 4$

Thus $3 / 4 \leq A \leq 1$.

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