Question

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

Answer 1

`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`