Now I think I have the correct answer, but am not 100% sure. Is it correct, and if not, where did I screw up? Cheers.

Question

Find a functionf(x)such that the point (1, 2) is on the graph of y =f(x), the slope of the tangent line at (1, 2) is 3 andf’’(x)=x– 1.

My answer:

Equation of slope of the tangent at (1, 2) isy= 3x

f’’(x)=x– 1

thus:

f’(x)= ½x² –x+ c

To find c, we need to solve:

f’(x)= ½x² –x+ c = 3 atx= 1

c = 3½

Now we can findf(x):

f(x)=x³/6 – ½x²+ 3½x+ c

At the tangent point (x= 1), y=f(x)has to touch the curve y = 3x:

c = 1/6

f(x)=x³/6 – ½x²+ 3½x+ 1/6