Question

The minimum value of $f\left(x\right)=si{n}^{4}x+co{s}^{4}x,0\le x\le \frac{\pi }{2}$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

$f\left(x\right)=si{n}^{4}x+co{s}^{4}x={(si{n}^{2}x+co{s}^2)}^2-2si{n}^{2}xco{s}^{2}x\Rightarrow f\left(x\right)=1-\frac{1}{2}si{n}^{2}2x\because 0\le si{n}^{2}2x\le 1\Rightarrow 0\le 2-2f\left(x\right)\le 1\Rightarrow -2\le -2f\left(x\right)\le -1\Rightarrow \frac{1}{2}\le f\left(x\right)\le 1\therefore Minimumvalueoff\left(x\right)is\frac{1}{2}$