continuity, differentiability and the maximum of two numbers
Hi, could anyone offer some help with the following? I am not sure how to start it.
for real numbers x and y let x v y be the maximum of x and y. For functions f,g (from the reals to the reals) define the function f v g as follows: (f v g)(x)=f(x) v g(x).
prove the following:
a. x v y = 0.5(|x-y| +x+y)
b. If functions f, g are continuous then so is f v g.
c. If f(0)=g(0) and f and g both have a local maximum at 0 then f v g has a minimum at 0.
d. If f(x) tends to infinity and g(x) tends to negative infinity as x tends to infinity then f v g tends to infinity as x tends to infinity.
e. If f and g are continuous and f(0)>g(0), g(1)>f(1) then there exists c in (0,1) with f(c)=g(c). If also f,g and f v g are differentiable the f'(c)=g'(c) for all such c.