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Proof using Mean Value Theorem of Continuous, Increasing Functions

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.

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A proof using Mean Value Theorem of Continuous, Increasing Functions is provided. The solution is detailed and well presented.

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