Find-maximum-and-minimum-value-of-function-y-cos-2-x-cos-x-3-on-interval-0-x-pi-2-

Question Number 133782 by bramlexs22 last updated on 24/Feb/21

Find maximum and minimum

value of function y = \cos^{2} x + \cos x + 3

on interval 0 ⩽ x ⩽ \frac{π}{2}

Answered by bobhans last updated on 24/Feb/21

y = {(\cos x + \frac{1}{2})}^{2} + \frac{11}{4}

w e k n o w 0 ⩽ \cos x ⩽ 1 o n i n t e r v a l 0 ⩽ x ⩽ \frac{π}{2}

s o \frac{1}{2} ⩽ \cos x + \frac{1}{2} ⩽ \frac{3}{2} a n d

\frac{1}{4} ⩽ {(\cos x + \frac{1}{2})}^{2} ⩽ \frac{9}{4}; t h u s

\frac{1}{4} + \frac{11}{4} ⩽ {(\cos x + \frac{1}{2})}^{2} + \frac{11}{4} ⩽ \frac{9}{4} + \frac{11}{4}

\Leftrightarrow 3 ⩽ y ⩽ 5 \Rightarrow {\begin{cases} m a x v a l u e = 5 \\ m i n v a l u e = 3 \end{cases}

Commented by bramlexs22 last updated on 24/Feb/21