If-y-cos-2-x-sin-4-x-for-all-values-of-x-then-prove-that-3-4-y-1-

Question Number 161396 by abdullah_ff last updated on 17/Dec/21

If y = {c o s}^{2} x + {s i n}^{4} x for all values of x,

then prove that

\frac{3}{4} \leq y \leq 1

Commented by cortano last updated on 17/Dec/21

y = \cos^{2} x + {(1 - \cos^{2} x)}^{2}

y = \cos^{4} x - \cos^{2} x + 1

y = {(\cos^{2} x - \frac{1}{2})}^{2} + \frac{3}{4}

\Rightarrow 0 ⩽ \cos^{2} x ⩽ 1

{\begin{cases} m i n = \frac{3}{4} w h e n \cos^{2} x = \frac{1}{4} \\ m a x = 1 w h e n \cos^{2} x = 1 \end{cases}

Commented by abdullah_ff last updated on 17/Dec/21

t h a n k y o u m y d e a r s i r . .