Question

State Rolle’s theorem. Using Rolle’s theorem, find a point on the curve

f(x) = cos2x where the tangent is parallel to x-axis on [−π, π].

Answer 1

Rolle's theorem:

If a function
`f` is continuous on the closed interval
`[a, b]` and differentiable on the open interval
`(a,b)` such that
`f(a) = f(b)`, then [tex)f′(x) = 0[/tex] for some
`x` with
`a \leq x \leq b`.

Tangent:

A tangent is parallel to the x-axis if the slope of the tangent is 0. So, using Rolle's theorem, if we consider
`x=-\pi/2` and
`x=\pi/2`, then
`f(-\pi/2)=f(\pi/2)=-1`.

Since
`f(x)` is continuous on
`[-\pi/2, \pi/2]` and differentiable on
`(-\pi/2, \pi/2)`,
`f'(x)=0` for some
`-\pi/2 \leq x \leq \pi/2`, and thus there is a point in this interval which is parallel to the x-axis.