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I implore your help. Thank you for asking questions in the mid-term exam of senior two mathematics.
The derived function f f' (x) = 3a * x 2+2b * x+c+c.

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.