Example $1$: A ladder $25$ feet long is leaning against a wall. The bottom of the ladder is being pulled away from the wall at $2f\frac{t}{sec}.$ How fast is the top of the ladder sliding down the wall when the top of the ladder is $7$ feet off of the ground.

Determine if Rolle's Theorem can be applied to $f$ on the indicated interval. If Rolle's Theorem can be applied, state all values of $c$ that it guarantees.

$f(x)=x^2-2x,[0,2]$

$f(x)=\frac{x^2-1}{x},[-1,1]$

$f(x)=x-x^{\frac{1}{3}},[0,1]$