The function f(x)= (x^3) + (ax^2) +bx + 7 has a relative extrema at x=1 and x=-3. Find a and b.

Answer 1

f'(x) = 3x²+2ax+b = 0 for x = -3, 1.

Substitute x with -3 and 1 and rewrite the equations as b in terms of a.

3(-3)²+2(a)(-3)+b = 0

27 - 6a + b = 0

b = 6a - 27

3(1)²+2(a)(1)+b = 0

3 + 2a + b = 0

b = -2a - 3

Set both equations for b equal to each other and solve for a.

6a - 27 = -2a - 3

8a = 24

a = 3

Substitute the value for a into one of the equations for b.

b = -2(3) - 3

b = -9