i am given two differentiable function f and g .
prove that for u(x)=max(f(x),g(x))
and v(x)=min(f(x),g(x))
there is one sided derivatives
??
i suppose it is the derivative found by taking either the right- or left-hand limit.
that is:
$\displaystyle f'(x) = \lim_{h \to 0} \frac {f(x + h) - f(x)}h$ (the derivative)
$\displaystyle f'(x)_+ = \lim_{h \to 0^+} \frac {f(x + h) - f(x)}h$ (one-sided derivative...from the right)
$\displaystyle f'(x)_- = \lim_{h \to 0^-} \frac {f(x + h) - f(x)}h$ (one-sided derivative... from the left)
deffentiable function is a function which on soome point its
left side derivative equals its write side derivative.
so
if both functions are derivatable.
then it doent matter what function we take
there for sure will be a function where
it has a point for which there is a derivative on both sides
even though we need only one.
so its an over kill
am i correct?