According to the meaning of the question, x= 1, x=4 is the solution of the derivative function =0 equation, so: 3a+2b+c=0, 48a+8b+c=0.

Substituting x=2 into the derivative function gives f'(2)= 12a+4b+c=2 (the vertical slope is negative reciprocal).

Three equations are solved simultaneously: a=- 1/3, b=5/2 and c=-4.