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this is the increasing function theorem, hope it helps
"If F'(x) >= 0 , and all x's are and element of [a,b], Then F is increasing on [a,b]"
use Mean Value Theorem (M.V.T)
Let F'(x)>=0 on some interval
Let x1< x2 (points from that interval)
by M.V.T there is a point C which is an element of [x1,x2] such that
F(x2)-F(x1) / X2- X1 = F'(C)
this implies: F(x2)-F(x1) = F'(C) X [x2-x1]
F'(C)>=0
[x2-x1]>0
therefore: F(x2)>=F(x1)
Therefore: F is increasing on that interval.
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