Step 4 To find (f-l)'(x) use the theorem that states that if f is differentiable on an interval I and f has an inverse 1 function g, that is g = f-1, then g is differentiable at any x for which f '(g(x)) ± 0 and g'(x) %3D f'(g(x))' Find the values of f-1(121) and then f'(f-1(121)), f-1(x) = (x + 4)1/3 %3D f-1(121) %3D f'(F-1(121)) = f'I 15 = 31 15 75 75 %3D Step 5 1 Substitute the above values in the expression for (f-1)'(121): f'(F=1(121))" 1 (F-1)'(121) = f'(F=1(121)) 1 75 Thus, (f-1)'(121) = 5

Let F(x) = f(f(x)) and G(x) = (F(x))'. You also know that f(9) = 2, f(2) 3, f'(2) = 11, f'(9) – 9 Find F'(9) and G'(9) = Question Help: Video Submit Question

Let f(x) be a function that is differentiable for all x. Let g(x) be defined by g(x) = f(x) + f(3 – x). Show that g'(x) has a root in the interval (0,3). (a) (b) Let f(x) be continuous and differentiable on the interval [-8,0]. Suppose f(-8) = -2 and f'(x) < 3 for all x. What is the largest possible value for f (0)? Justify your answer.