Find-maximum-value-of-function-x-2x-16-x-35-x-

Question Number 151685 by iloveisrael last updated on 22/Aug/21

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

α (x) = \sqrt{2 x} + \sqrt{16 - x} + \sqrt{35 + x} .

Answered by mr W last updated on 22/Aug/21

x ⩾ 0

x ⩽ 16

f^{'} (x) = \frac{1}{\sqrt{2 x}} - \frac{1}{2 \sqrt{16 - x}} + \frac{1}{2 \sqrt{35 + x}} = 0

\frac{2}{\sqrt{2 x}} + \frac{1}{\sqrt{35 + x}} = \frac{1}{\sqrt{16 - x}}

\frac{2}{x} + \frac{1}{35 + x} + \frac{2 \sqrt{2}}{\sqrt{x (35 + x)}} = \frac{1}{16 - x}

\frac{2 \sqrt{2}}{\sqrt{x (35 + x)}} = \frac{x (35 + x) - x (16 - x) - 2 (35 + x) (16 - x)}{(16 - x) (x) (35 + x)}

\frac{2 \sqrt{2}}{\sqrt{x (35 + x)}} = \frac{4 x^{2} + 57 x - 1120}{(16 - x) (x) (35 + x)}

8 = \frac{{(4 x^{2} + 57 x - 1120)}^{2}}{{(16 - x)}^{2} (x) (35 + x)}

8 x^{4} + 432 x^{3} + 1201 x^{2} - 199360 x + 1254400 = 0

\Rightarrow x \approx 12.6106

f {(x)}_{m a x} \approx 13.7631

Commented by mr W last updated on 22/Aug/21

f {(x)}_{m i n} = f (0) = 4 + \sqrt{35}

Commented by MJS_new last updated on 22/Aug/21

I also tried . no useable exact solution possible

Commented by mr W last updated on 22/Aug/21

s o i t i s, s i r . i f o u n d n o e x a c t w a y .

Commented by iloveisrael last updated on 22/Aug/21

f o r m i n i m u m v a l u e, i g o t e x a c t v a l u e