The-maximum-value-of-the-expression-sin-2-x-2a-2-2a-2-3-cos-2-x-where-a-and-x-are-real-numbers-is-1-4-2-2-3-2-4-0-

Question Number 16090 by Tinkutara last updated on 17/Jun/17

The maximum value of the expression

∣ \sqrt{\sin^{2} x + 2 a^{2}} - \sqrt{2 a^{2} - 3 - \cos^{2} x} ∣;

where^{'} a^{'} and^{'} x^{'} are real numbers, is

(1) 4

(2) 2

(3) \sqrt{2}

(4) 0

Answered by ajfour last updated on 18/Jun/17

f o r m a x i m u m v a l u e l e t s i n x = 1,

\cos x = 0

t h e n m a x i m u m o f

f (z) = \sqrt{1 + 2 z^{2}} - \sqrt{2 z^{2} - 3}

i s a n e v e n f u n c t i o n w i t h

f^{'} (z) < 0 f o r z > \sqrt{3 / 2}

h e n c e m a x i m u m w h e n 2 z^{2} = 3

a n d m a x i m u m v a l u e o f

e x p r e s s i o n = 2 .

Commented by Tinkutara last updated on 18/Jun/17

Why f (z) is an even function with

f^{'} (z) < 0 ?