Find f if f ''(x) = 12x2 + 6x − 4, f(0) = 9, and f(1) = 1.

  f(x) = x^4 +x^3 -2x^2 -8x +9

Step-by-step explanation:

You know that the anitderivative of ax^b is ax^(b+1)/(b+1). The first antiderivative is ...

  f'(x) = 4x^3 +3x^2 -4x +p . . . . . where p is some constant

The second antiderivative is ...

  f(x) = x^4 +x^3 -2x^2 +px +q . . . . where q is also some constant

Then the constants can be found from ...

  f(0) = q = 9

  f(1) = 1 + 1 - 2 +p + 9 = 1

  p = -8

The solution is ...

  f(x) = x^4 +x^3 -2x^2 -8x +9

_____

The graphs verify the results. The second derivative is plotted against the given quadratic, and they are seen to overlap. The function values at x=0 and x=1 are the ones specified by the problem.