Question

Find constants a, b, and c such that the function f(x)=ax^3+bx^2+c will have a local extremum at (2,11) and a point of inflection at (1,5)

The answers are a=-3, b=9, and c=-1 but I can't get those.

I tried finding the first and second derivatives of the given function to find a and b when the derivatives equal zero (f'(2) and f''(1)) but no luck.

Answer 1

f' = 3 a x^2 + 2 b x = x (3 a x - 2 b)
= 0 when x = 0 and when x = (2/3)(b/a)
if x = 2 and y = 11
11 = 8 a + 4 b + c
but x = 2 = (2/3) b/a
b /a = 3 or b = 3 a so 11 = 8 a + 12 a + c = 20 a + c = 11
also = 0 at (1,5)
5 = a + b + c

second derivative = 0 at (1,5)
0 = 6 a x + 2 b
b = -3 a
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5 = a -3 a + c = -2 a + c
and
11 = 20 a + c
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-6 = -22 a etc (check my work , did it fast)
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