Max-amp-min-value-of-function-f-x-6-x-12-x-

Question Number 153956 by liberty last updated on 12/Sep/21

M a x & m i n v a l u e o f f u n c t i o n

f (x) = \sqrt{6 - x} + \sqrt{12 + x} .

Answered by EDWIN88 last updated on 12/Sep/21

d o m a i n o f f (x) = [- 12, 6]

b y C h a u c y - S c h w a r d i n e q u a l i t y

2 [(6 - x) + (12 + x)] ⩾ {(\sqrt{6 - x} + \sqrt{12 + x})}^{2}

2 \times 18 ⩾ {[\sqrt{6 - x} + \sqrt{12 + x}]}^{2}

\Rightarrow \sqrt{18} ⩽ \sqrt{6 - x} + \sqrt{12 + x} ⩽ \sqrt{36}

\Rightarrow 3 \sqrt{2} ⩽ \sqrt{6 - x} + \sqrt{12 + x} ⩽ 6

{\begin{cases} m i n = 3 \sqrt{2} \\ m a x = 6 \end{cases}

Commented by liberty last updated on 13/Sep/21

y e s c o r r e c t

Answered by ajfour last updated on 15/Sep/21

l e t x - 3 = t

f (t) = \sqrt{9 - t} + \sqrt{9 + t}

\Rightarrow t \in [- 9, 9]

f^{2} (t) = 18 + 2 \sqrt{81 - t^{2}}

c l e a r l y f_{m i n} = 3 \sqrt{2}, f_{m a x} = 6

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