for-x-gt-0-find-the-minimum-of-the-function-f-x-x-3-5-x-

Question Number 194837 by mr W last updated on 16/Jul/23

f o r x > 0 f i n d t h e m i n i m u m o f t h e

f u n c t i o n f (x) = x^{3} + \frac{5}{x} .

Answered by Frix last updated on 16/Jul/23

f^{'} (x) = 0

3 x^{2} - \frac{5}{x^{2}} = 0

x = \pm {(\frac{5}{3})}^{\frac{1}{4}}

f ({(\frac{5}{3})}^{\frac{1}{4}}) = 4 \times {(\frac{5}{3})}^{\frac{3}{4}}

Answered by mr W last updated on 16/Jul/23

a n o t h e r w a y :

f (x) = x^{3} + \frac{5}{3 x} + \frac{5}{3 x} + \frac{5}{3 x}

⩾ 4 {(x^{3} \times \frac{5}{3 x} \times \frac{5}{3 x} \times \frac{5}{3 x})}^{\frac{1}{4}} = 4 {(\frac{5}{3})}^{\frac{3}{4}}

\Rightarrow m i n i m u m = 4 {(\frac{5}{3})}^{\frac{3}{4}}

w h e n x^{3} = \frac{5}{3 x}, i . e . x = {(\frac{5}{3})}^{\frac{1}{4}}

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