Mean Value Theorem

Question

Let f be the function given by f(x)= (x²-9)/sinx on the closed interval [0,5]. Of the following intervals on which can the MVT be applied to f?

I. [1,3] because f is continuous on [1,3] and differentiable on (1,3).

II. [4,5] because f is continuous on [4,5] and differentiable on (4,5).

III. [1,4] because f is continuous on [1,4] and differentiable on (1,4).

Select all that apply.

Answer 1

MVT

If f is cont on [a,b] and differentiable on (a,b), then there is at least one c in (a,b) such that

f'(c) = (f(b) - f(a)) / (b - a)

1. sinx cannot be 0 in order to apply MVT

1. works out
2. works out
3. does not work. pi ~~ 3.14 => x = pi, f is not continuous, so MVT fails.