minimum-value-of-x-6-2-25-x-6-2-121-equal-to-

Question Number 135925 by liberty last updated on 17/Mar/21

m i n i m u m v a l u e o f \sqrt{{(x + 6)}^{2} + 25} + \sqrt{{(x - 6)}^{2} + 121}

e q u a l t o ?

Answered by john_santu last updated on 17/Mar/21

M i n i m u m v a l u e o f

\sqrt{{(x + 6)}^{2} + 25} + \sqrt{{(x - 6)}^{2} + 121} e q u a l t o

s t a n d a r t m e t h o d

f (x) = \sqrt{{(x + 6)}^{2} + 25} + \sqrt{{(x - 6)}^{2} + 121}

f^{'} (x) = \frac{x + 6}{\sqrt{{(x + 6)}^{2} + 25}} + \frac{x - 6}{\sqrt{{(x - 6)}^{2} + 121}} = 0

\frac{x + 6}{\sqrt{{(x + 6)}^{2} + 25}} = \frac{6 - x}{\sqrt{{(x - 6)}^{2} + 121}}

\Rightarrow \frac{{(x + 6)}^{2}}{{(x + 6)}^{2} + 25} = \frac{{(6 - x)}^{2}}{{(6 - x)}^{2} + 121}

\Rightarrow \frac{{(x + 6)}^{2} + 25}{{(x + 6)}^{2}} = \frac{{(6 - x)}^{2} + 121}{{(6 - x)}^{2}}

\Rightarrow {(\frac{5}{x + 6})}^{2} = {(\frac{11}{6 - x})}^{2}

w e g e t {\begin{cases} \frac{5}{x + 6} = \frac{11}{6 - x} \\ \frac{5}{x + 6} = \frac{11}{x - 6} \end{cases}

(1) 30 - 5 x = 11 x + 66, 16 x = - 36, x = - \frac{9}{4}

(2) 5 x - 30 = 11 x + 66, 6 x = - 96, x = - 16

f o r x = - \frac{9}{4} \Rightarrow f_{1} = \sqrt{{(\frac{15}{4})}^{2} + 25} + \sqrt{{(- \frac{33}{4})}^{2} + 121}

= 20 (m i n i m u m v a l u e)

f o r x = - 16 \Rightarrow f_{2} = \sqrt{{(- 10)}^{2} + 25} + \sqrt{{(- 22)}^{2} + 121}

= 16 \sqrt{5} \approx 35.77 (m a x i m u m v a l u e)

Commented by mr W last updated on 17/Mar/21

m a x i m u m {d o e s n}^{'} t e x i s t, s i n c e

\lim_{x \to - \infty} f (x) = + \infty

\lim_{x \to + \infty} f (x) = + \infty

Answered by mr W last updated on 17/Mar/21

Commented by mr W last updated on 17/Mar/21

w e c a n s o l v e t h i s p r o b l e m w i t h o u t

u s i n g c a l c u l u s .

O A = \sqrt{{(x + 6)}^{2} + 5^{2}}

O B = \sqrt{{(x - 6)}^{2} + 11^{2}}

m i n i n u m f r o m O A + O B i s t h e

s t r a i g h t l i n e A B = \sqrt{{(6 + 6)}^{2} + {(5 + 11)}^{2}}

= 20, w h e n t h e o r i g i n i s a t p o i n t C, o r

x = - \frac{9}{4} .

Commented by mr W last updated on 17/Mar/21

g e n e r a l l y u s i n g t h i s w e g e t

m i n i n u m f r o m

\sqrt{{(x + a)}^{2} + c^{2}} + \sqrt{{(x + b)}^{2} + d^{2}} i s

\sqrt{{(a - b)}^{2} + {(c + d)}^{2}}

(a s s u m e c, d ⩾ 0)

Commented by liberty last updated on 17/Mar/21

w a w \dots i t f o l l o w s t h a t i f

\sqrt{{(x + 3)}^{2} + 36} + \sqrt{{(x - 3)}^{2} + 49}

t h e m e t h o d v a l i d s i r ?

Commented by mr W last updated on 17/Mar/21

y e s .

a n o t h e r e x a m p l e :

{[\sqrt{{(x + 3)}^{2} + 10} + \sqrt{{(x + 8)}^{2} + 20}]}_{m i n}

= \sqrt{{(8 - 3)}^{2} + {(\sqrt{10} + \sqrt{20})}^{2}} = \sqrt{55 + 20 \sqrt{2}}

y o u c a n c h e c k i t w i t h G r a p h e r o r

u s i n g c a l c u l u s .

Commented by liberty last updated on 17/Mar/21

g r e a t \dots .

Commented by Dwaipayan Shikari last updated on 17/Mar/21

N e w t o n u s e d t o h a v e G e o m e t r i c a l M e t h o d o f C a l c u l u s