f-x-x-a-3a-x-with-a-gt-0-is-given-If-f-max-f-min-32-find-a-

Question Number 186265 by mnjuly1970 last updated on 02/Feb/23

f (x) = \sqrt{x - a} + \sqrt{3 a - x} w i t h (a > 0)

i s g i v e n . I f, f_{m a x} . f_{m i n} = \sqrt{32}

f i n d,^{″} a^{″} = ?

Answered by mahdipoor last updated on 02/Feb/23

\frac{d f}{d x} = \frac{1}{2 \sqrt{x - a}} + \frac{- 1}{2 \sqrt{3 a - x}} = 0 \Rightarrow \sqrt{3 a - x} = \sqrt{x - a}

\Rightarrow x = 2 a

D_{f} = [a, 3 a]

f (a) = f (3 a) = \sqrt{2 a} f (2 a) = 2 \sqrt{a}

f_{m i n} . f_{m a x} = \sqrt{8 a^{2}} = \sqrt{32} \Rightarrow a = 2

w a y 2 :

f^{2} = 2 a + 2 \sqrt{4 a x - x^{2} - 3 a^{2}}

g (x) = \sqrt{4 a x - x^{2} - 3 a^{2}} {\begin{cases} m i n = 0 \\ m a x = g (\frac{- 4 a}{- 1 \times 2}) = a \end{cases}

m i n f^{2} \times m a x f^{2} = 2 a \times 4 a = {(\sqrt{32})}^{2} \Rightarrow a = 2