Let f(x) = tan()
Show that f() = f() but there is no number c in the
closed interval [,] such that f'(c) = 0.
f() = tan() = tan( = 0
f() = tan() = tan( = 0
f'(x) = secē()
The secant function is never 0, so there can be no value
of c such that f'(c) = 0 on that interval.
Why does this not contradict Rolle's Theorem?
Because Rolle's theorem only says there is such a number c on closed interval
[a,b] in which the function is everywhere defined, continuous and differentiable
on the closed interval [a,b]. Rolle's theorem is not violated since
f(x) = tan() is not continuous and differentiable everywhere on the
closed interval [,] for f(x) is not defined ar x =
which is on the closed interval [,].
Edwin