utf14-001
True or False. If false give a counter-example.
If $f(x)$ is continuous at $x=2$ and $f(2)$ is the maximum
y-value of the function then $f(x)$ is differentiable at $x=2$.
You take a boat trip from New York to London and follow a
smooth, but curvy course. At some point on your journey you are traveling
parallel to the direct straight line course.
On a trip from Dallas to Austin you go through Waco at 10 p.m. and
Temple at 11 p.m. (50 miles apart). Between 10 and 11, at some point, you
were driving exactly 50 $mph$.
If $f(x)$ is defined on $ [0,1] $ and continuous and differentiable
on $(0,1)$ then there exists a point $x_0$ in $ [0,1] $ such that
$f'(x_0) = f(1)-f(0).$