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Question
Let f(x)=x^3 +Ax^2 +Bx -15 be a function whose graph has a maximum at x=2 and a point of inflection at x=3. Find the values of A and B.
A. A=-6, B=12
B. A=-9, B=12
C. A=-9, B=24
D. A=9, B=-48
E. A=9, B=24

Answer
Hi Michelle,
f(x) = x + Ax + Bx - 15
f'(x) = 3x + 2Ax + B
f''(x) = 6x + 2A

For a maximum point the first derivative f'(x) is zero and the second derivative f''(x) is positive.
So, at x = 2
3(2) + 2A(2) + B = 0
12 + 4A + B = 0

A necessary condition for a point of inflection is that f''(x), if it exists, = 0.
So, at x = 3
6(3) + 2A = 0
18 + 2A = 0
A = -9

Going back to 12 + 4A + B = 0, we have
12 + 4(-9) + B = 0
12 - 36 + B = 0
B = 24

Regards

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