Find-minimum-value-of-function-f-x-sin-7-x-cos-11-x-

Question Number 124404 by benjo_mathlover last updated on 03/Dec/20

F i n d m i n i m u m v a l u e o f f u n c t i o n

f (x) = \sin^{7} (x) + \cos^{11} (x)

Commented by benjo_mathlover last updated on 03/Dec/20

Commented by benjo_mathlover last updated on 03/Dec/20

y e s . b u t h o w t h e w a y i n a l g e b r a i c ? ?

Commented by MJS_new last updated on 03/Dec/20

- 1 ⩽ f (x) ⩽ 1

is there an easy way to show ?

Answered by mindispower last updated on 03/Dec/20

{s i n}^{7} (x) = {s i n}^{2} (x) . {s i n}^{5} (x)

- 1 ⩽ {s i n}^{5} (x) ⩽ 1,

- 1 ⩽ {c o s}^{9} (x ⩽ 1

\Rightarrow - {s i n}^{2} (x) - {c o s}^{2} (x) ⩽ {s i n}^{7} (x) + {c o s}^{11} (x) ⩽ {s i n}^{2} (x) + {c o s}^{2} (x)

\Leftrightarrow

- 1 ⩽ f (x) ⩽ 1

f (0) = 1

f (- \frac{π}{2}) = - 1

m i n f = - 1, m a x f = 1