FInd the values of c

Question

find the value or values of c that satisfy the equation f(b)-f(a) / b-a=f'(c) in the conclusion of the Mean Value Theorem for the following function and interval. f(x)= 3x^3 + 4x^2, [-2, 2]

Answer 1

f'(x) = 9x² + 8x; f(-2) = - 24 + 16 = - 8; f(2) = 24 + 16 = 40;

f'(c) = 9c² + 8c = (40 + 8)/(2 + 2) = 12; 9c² + 8c - 12 = 0; c = (- 4 ±√16 + 108)/9 = (- 4 ± 2√31)/9

c₁ = - 1.866, c₂ = 0.793