Find y=f(x) if f''(x)=6x-2, f'(1)=3, and f(2)=7

Pedro Gaspar asked Jun 7, 2023

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Pedro Gaspar asked Jun 7, 2023

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13 votes

Answer:

f(x)=x^3-x^2+2x-1

Explanation:

Since
f''(x)=6x-2 , then
f'(x)=3x^2-2x+C , thus:

$f'(x)=3x^2-2x+C\\\\f'(1)=3(1)^2-2(1)+C\\\\3=3-2+C\\\\3=1+C\\\\2=C$

Therefore, since
f'(x)=3x^2-2x+2 , then
f(x)=x^3-x^2+2x+C , thus:

$f(x)=x^3-x^2+2x+C\\\\f(2)=2^3-2^2+2(2)+C\\\\7=8-4+4+C\\\\7=8+C\\\\-1=C$

In conclusion,
f(x)=x^3-x^2+2x-1

Shepherd answered Jun 12, 2023

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