The function f is given by f(x)=(x^3+bx+6)g(x), where b is constant and g is a differentiable function satisfying g(2)=3 and g'(2)= -1. For what value of b is f'(2)=0?

Question

The function f is given by f(x)=(x^3+bx+6)g(x), where b is constant and g is a differentiable function satisfying g(2)=3 and g'(2)= -1. For what value of b is f'(2)=0?

Answer 1

f'(x) = (3x² + b)g(x) + (x³ + bx + 6)g'(x); f'(2) = (12 + b)g(2) + (8 + 2b + 6)g'(2) = 3(12 + b) - 1(14 + 2b) = 0;

36 + 3b - 14 - 2b = 0; b = - 22