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f-x-2-4f-x-8x-2-32x-28-f-x-

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Question Number 205825 by tri26112004 last updated on 31/Mar/24
[f′(x)]^2 +4f(x)=8x^2 −32x+28 ⇒f(x)=¿
[f′(x)]2+4f(x)=8x2−32x+28⇒f(x)=¿
Commented by mr W last updated on 31/Mar/24
f(x)=x^2 −4x+3 or f(x)=−2x^2 +8x−9 or ...
f(x)=x2−4x+3orf(x)=−2x2+8x−9or…
Answered by MM42 last updated on 31/Mar/24
f=ax^2 +bx+c⇒f′=2ax+b ⇒4a^2 x^2 +4abx+b^2 +4ax^2 +4bx+4c=8x^2 −32x+28 ⇒4a^2 +4a=8⇒a=1 or a=−2 4ab+4b=−32 & b^2 +4c=28 if a=1⇒b=−4 & c=3 ⇒f(x)=x^2 −4x+3 ✓ if a=−2⇒b=8 & c=−9 ⇒f(x)=−2x^2 +8x−9 ✓
f=ax2+bx+c⇒f′=2ax+b⇒4a2x2+4abx+b2+4ax2+4bx+4c=8x2−32x+28⇒4a2+4a=8⇒a=1ora=−24ab+4b=−32&b2+4c=28ifa=1⇒b=−4&c=3⇒f(x)=x2−4x+3✓ifa=−2⇒b=8&c=−9⇒f(x)=−2x2+8x−9✓

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