Let f(x) = 3x^3 + 4e^ x 2 and g(x) = f⁻¹(x) . Then the value of g'(4) is equal to...

Step-by-step solution

Given f(x) = 3x^3 + 4e^ x 2 and g(x) = f⁻¹(x) . Using the derivative of an inverse function, g'(x) = 1 f'(g(x)) . For x = 4 , g'(4) = 1 f'(g(4)) . To find g(4) , we set f(x) = 4 : 3x^3 + 4e^ x 2 = 4 By inspection, x = 0 satisfies the equation, so g(4) = 0 . Differentiating f(x) with respect to x : f'(x) = 9x^2 + 4 1 2 e^ x 2 = 9x^2 + 2e^ x 2 Evaluating f'(x) at x = 0 : f'(0) = 9(0)^2 + 2e⁰ = 2 Therefore, g'(4) = 1 f'(0) = 1 2 . Answer: 1 2