find-minimum-value-of-x-2-4-x-2-24x-153-for-x-0-in-R-

Question Number 80027 by jagoll last updated on 30/Jan/20

f i n d m i n i m u m

v a l u e o f \sqrt{x^{2} + 4} + \sqrt{x^{2} - 24 x + 153}

f o r x ⩾ 0 i n R

Commented by john santu last updated on 30/Jan/20

yes sir

Commented by john santu last updated on 30/Jan/20

f^{'} (x) = \frac{x}{\sqrt{x^{2} + 4}} + \frac{x - 12}{\sqrt{x^{2} - 24 x + 153}} = 0

\frac{x}{12 - x} = \sqrt{\frac{x^{2} + 4}{x^{2} - 24 x + 153}}

\frac{x^{2}}{144 - 24 x + x^{2}} = \frac{x^{2} + x}{x^{2} - 24 x + 153}

x^{4} - {24 x}^{3} + {153 x}^{2} = x^{4} - {24 x}^{3} + {144 x}^{2}

+ {4 x}^{2} - 96 x + 576

{5 x}^{2} + 96 x - 576 = 0

x = \frac{- 96 + \sqrt{20736}}{10} = \frac{- 96 + 144}{10} = 4.8

f_{\min} = \sqrt{{(\frac{24}{5})}^{2} + 4} + \sqrt{{(\frac{24}{5} - 12)}^{2} + 9}

Commented by mr W last updated on 30/Jan/20

w i t h x = 4.8 w e g e t e x a c t l y f = 13 .

i . e . f_{m i n} = 13 .

Commented by jagoll last updated on 30/Jan/20

t h a n k s m r w a n d j o h n

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