Question

Given the function `f(x)= 6 cos (x)`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-pi/2, pi/2] and find the c?

Answer 1

The Mean Value Theorem has two hypotheses:

H1 : `f` is continuous on the closed interval `[a,b]`

H2 : `f` is differentiable on the open interval `(a,b)`.

`f` satisfies the hypotheses on `[a,b]` if and only if the hypotheses are true for `f` on `[a,b]` (and on `(a,b)` for H2.)

So you need to determine whether `f(x) = 6cosx` is continuous on `[-pi/2,pi/2]` and differentiable on `(-pi/2,pi/2)`.

(It is)

The conclusion on the Mean Value Theorem is

C : there is a `c` in `(a,b)` such that `f'(c) = (f(b)-f(a))/(b-a)`

To find the `c` mentioned in the conclusion, you need to solve the equation. Discard any solutions outside of `(a,b) = (-pi/2,pi/2)`.

(`c=0`)