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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
Findx,suchthatf(x)isminimum.f(x)={c2−x2c−x−(c−x)}2
Commented by ajfour last updated on 17/Jan/22
did.
did.
Commented by MJS_new last updated on 17/Jan/22
no x on the rhs... please correct
noxontherhs…pleasecorrect
Answered by MJS_new last updated on 17/Jan/22
(√(f(x)))=0 has one solution for c∈R ⇒ min(f(x))=0 at f(x)=0 ⇒ ((√(c^2 −x^2 ))/(c−x))−(c−x)=0 ⇔ x^3 −3cx^2 +(3c^2 +1)x−c(c^2 −1)=0 ⇒ x=c+((−c+(√((27c^2 +1)/(27)))))^(1/3) −((c+(√((27c^2 +1)/(27)))))^(1/3)
f(x)=0hasonesolutionforc∈R⇒min(f(x))=0atf(x)=0⇒c2−x2c−x−(c−x)=0⇔x3−3cx2+(3c2+1)x−c(c2−1)=0⇒x=c+−c+27c2+1273−c+27c2+1273
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)={c2−x2c−x−(c−x)}2⩾0f(x)min=0whenc2−x2c−x−(c−x)=0c2−x2=(c−x)2c+x=(c−x)3(c−x)3+(c−x)−2c=0lett=c−xt3+t−2c=0t=c2+127+c3−c2+127−c3⇒x=c−c2+127+c3+c2+127−c3

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