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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

Hi Michelle,

f(x) = x³ + Ax² + Bx - 15

f'(x) = 3x² + 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)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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