Find x, such that f(x) is minimum. f(x)={((√(c^2 −x^2 ))/(c−x))−(c−x)}^2


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Question Number 164462 by ajfour last updated on 17/Jan/22
$Find x, such that f(x) is minimum. f(x)={((√(c^2 −x^2 ))/(c−x))−(c−x)}^2$
$F i n d x, s u c h t h a t f (x) i s m i n i m u m .$ $f (x) = {\frac{\sqrt{c^{2} - x^{2}}}{c - x} - (c - x)}^{2}$
Commented by ajfour last updated on 17/Jan/22

$d i d .$
Commented by MJS_new last updated on 17/Jan/22

$no x on the rhs . . . please correct$
	Answered by MJS_new last updated on 17/Jan/22

	$\sqrt{f (x)} = 0 has one solution for c \in R$ $\Rightarrow$ $\min (f (x)) = 0 at f (x) = 0$ $\Rightarrow$ $\frac{\sqrt{c^{2} - x^{2}}}{c - x} - (c - x) = 0$ $\Leftrightarrow$ $x^{3} - 3 {c x}^{2} + (3 c^{2} + 1) x - c (c^{2} - 1) = 0$ $\Rightarrow$ $x = c + \sqrt[3]{- c + \sqrt{\frac{27 c^{2} + 1}{27}}} - \sqrt[3]{c + \sqrt{\frac{27 c^{2} + 1}{27}}}$
	Answered by mr W last updated on 17/Jan/22
	$f(x)={((√(c^2 −x^2 ))/(c−x))−(c−x)}^2 ≥0 f(x)_(min) =0 when ((√(c^2 −x^2 ))/(c−x))−(c−x)=0 (√(c^2 −x^2 ))=(c−x)^2 c+x=(c−x)^3 (c−x)^3 +(c−x)−2c=0 let t=c−x t^3 +t−2c=0 t=(((√(c^2 +(1/(27))))+c))^(1/3) −(((√(c^2 +(1/(27))))−c))^(1/3) ⇒x=c−(((√(c^2 +(1/(27))))+c))^(1/3) +(((√(c^2 +(1/(27))))−c))^(1/3)$
	$f (x) = {\frac{\sqrt{c^{2} - x^{2}}}{c - x} - (c - x)}^{2} ⩾ 0$ $f {(x)}_{m i n} = 0 w h e n$ $\frac{\sqrt{c^{2} - x^{2}}}{c - x} - (c - x) = 0$ $\sqrt{c^{2} - x^{2}} = {(c - x)}^{2}$ $c + x = {(c - x)}^{3}$ ${(c - x)}^{3} + (c - x) - 2 c = 0$ $l e t t = c - x$ $t^{3} + t - 2 c = 0$ $t = \sqrt[3]{\sqrt{c^{2} + \frac{1}{27}} + c} - \sqrt[3]{\sqrt{c^{2} + \frac{1}{27}} - c}$ $\Rightarrow x = c - \sqrt[3]{\sqrt{c^{2} + \frac{1}{27}} + c} + \sqrt[3]{\sqrt{c^{2} + \frac{1}{27}} - c}$
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