Answer to Question #155740 in Calculus for Tom Garland

Question

Selected values of the function f are shown in the table below. the function f is continuous on the closed interval [-2,2] and differentiable on the open interval (-2,2). Determine the validity of the following statement: there exists a value x in the open interval (-2,2) such that f'(x)=0. x: -2, -1, 2 and f(x): -10, -13, -10

Answer 1

By Rolle's theorem, ( a special case of Mean Value Theorem),

i) The function f is continous on the closed interval [-2, 2]

ii)The function f is differentiable on the open interval (-2, 2)

iii)The value of f at the endpoints are the same, i.e. f(-2)=f(2)=-10

Then there exists a value x in the open interval (-2, 2) such that f'(x)=0.