Assume that f and f ' are continuous on [a,b], and f ''(x) exists and f ''(x)>0 for each x
1) Prove that f ' is increasing on [a,b]
Hint: the graph is concave up on this interval.
2) Prove that f(x) f(c) for each x if c and f '(c)=0.
A proof using Mean Value Theorem of Continuous, Increasing Functions is provided. The solution is detailed and well presented.