Question

${\sin }^{2}x+{\cos }^{4}x\in$

• A. $\left[\frac{1}{4}\frac{1}{2}\right]$
• B. $\left[\frac{3}{4},1\right]$
• C. $[-1,-\frac{3}{4}]$
• D. $[-\frac{3}{4},\frac{3}{4}]$

Answer 1

Answer: (B)

$\left[\frac{3}{4},1\right]$

Let $f\left(x\right)={\sin }^{2}x+{\cos }^{4}x$
$={\cos }^{4}x-{\cos }^{2}x+1[co{s}^{2}x+si{n}^{2}x=1]$
$={\cos }^{2}x({\cos }^{2}x-1)+1$
$=1-{\sin }^{2}x{\cos }^{2}x$
$f\left(x\right)=1-\frac{(\sin 2x{)}^2}{4}[\sin 2x=2\sin x\cos x]$
range of $f\left(x\right)ϵ[1-\frac{1}{4},1-\frac{0}{4}]$
$ϵ[\frac{3}{4},1]$