Find f if f ''(x) = 12x2 + 6x − 4, f(0) = 5, and f(1) = 4.

This is a differential equations problem.  We are to work backwards and determine the function f(x) when given f "(x) and initial values.

f ''(x) = 12x^2 + 6x − 4, when integrated with respect to x, yields:

                x^3        x^2
f '(x) = 12 + 6 - 4x + C, or 4x^3 + 3x^2 - 4x + C, and
                  3            2

             x^4         x^3        x^2
f(x) = 4 + 3 - 4 + Cx + D, or f(x)=x^4 + x^3 - 2x^2 + Cx + D
               4              3         2

Now, because f(0)=5, 5=0^4 + 0^3 -2(0)^2 + C(0) + D, so that D=5.
Determine D in the same manner:  Let x=1 and find the value of C.

Then the solution, f(x), is x^4 + x^3 - 2x^2 + Cx + 5.  Replace C with this value and then you'll have the desired function f(x).

The answer is