Let f : R R be a function defined by f(x) = x^5 + x^3 and let g(x) = f⁻¹(x) be the inverse of f . If g''(2) = a b where a and b are positive coprime integers, then the value of a is

Step-by-step solution

Given f(x) = x^5 + x^3 and g(x) = f⁻¹(x) . By the definition of inverse functions, we have f(g(x)) = x . Differentiating both sides with respect to x , we get: f'(g(x)) g'(x) = 1 g'(x) = 1 f'(g(x)) Differentiating again with respect to x using the chain rule and quotient rule: g''(x) = - f''(g(x)) g'(x) (f'(g(x)))^2 Substituting g'(x) = 1 f'(g(x)) into the equation: g''(x) = - f''(g(x)) (f'(g(x)))^3 To find g''(2) , we first determine g(2) . Since f(1) = 1^5 + 1^3 = 2 , it follows that g(2) = 1 . Now, we find the f