Is this right? Finding function f(x) given tangent slope and f''(x)

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 function *f(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 and *f’’(x)*= *x* – 1.

**My answer:**

Equation of slope of the tangent at (1, 2) is *y*= 3*x*

*f’’(x)*= *x* – 1

thus:

*f’(x**)*= ½*x*² – *x *+ c

To find c, we need to solve:

*f’(x)*= ½*x*² – *x *+ c = 3 at *x* = 1

c = 3½

Now we can find *f(x)*:

*f(x)*= *x*³/6 – ½*x*²+ 3½*x* + c

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

c = 1/6

*f(x)*= *x*³/6 – ½*x*²+ 3½*x* + 1/6