Lemma: Derivative of Increasing and Decreasing Functions
Let f be a continuous function on (a,b) where the at no point f'(x)=0.
If for all x on [a,b], f(x) is increasing, then f'(x) is positive.
If for all x on [a,b], f(x) is decreasing, then f'(x) is negative.
Proof:
(1) From the definition of derivatives (see Definition 1, here):
f'(x) = lim (Δx → 0) [f(x + Δx) - f(x)]/(Δx)
So, the sign of f'(x) is the sign of [f(x + Δx) - f(x)]/Δx and we can assume that is is nonzero.
(2) Case I: Δx is positive
If f(x) is strictly increasing, then f(x + Δx) - f(x) is positive and f'(x) is positive.
If f(x) is striclty decreasing, then f(x + Δx) - f(x) is negative and f'(x) is negative.
(3) Case II: Δx is negative
If f(x) is strictly increasing, then f(x + Δx) - f(x) is negative and f'(x) is positive
If f(x) is strictly decreasing, then f(x + Δx) - f(x) is positive and f'(x) is negative.
QED
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