Question

How do you find the number c that satisfies the conclusion of Rolle's Theorem for `f(x) = cos(5x)` for [π/20,7π/20]?

Answer 1

Solve `-5sin(5x) = 0` with `pi/20 < x < (7 pi)/20`

Explanation:

the conclusion of Rolle's Theorem is that there is a `c` in the interior of the interval under consideration at which `f'(c) =0`

For `f(x) = cos(5x)`, we have `f'(x) = -5sin(5x)`

We need to solve `-5sin(5x) = 0` in the interval `( pi/20, (7pi)/20 )` (That is, with `pi/20 < x < (7 pi)/20`)

`sin(5x) = 0` when `5x = 0 + kpi = k pi` for integer `k`.

Or at `x = 1/5 k pi = 4/20 k pi` for integer `k`.

Clearly, the only solution in the interval occurs when we take `k = 1` and get `x = pi / 5`