Tuesday, 24 April 2018
2:33 PM
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As you know from school, the sign of f' (x) will tell us whether f (x) is increasing
or decreasing. To carefully prove the relevant facts we use the MVT
Theorem
Let f be continuous on [a, b] and differentiable on (a, b).
o If f' (x) > 0 for all in (a, b) then f is increasing on [a, b].
If f'(a:) < 0 for all in (a, b) then f is decreasing on [a, b].
If f = 0 for all in (a, b) then f is constant on [a, b].
Proof. Suppose that f' (r) > 0 for all in (a, b) and choose two points and
in [a, b] such that <
Since f is differentiable on (a, b), it is continuous on [11, x2] and differentiable on
Hence, by the MVT,
f 02) — f (Xl)
for some c in (Xl, x2).
Accordingly,
f(X2) — f(X1) = — Xl) > 0.
The remaining two statements are proven in a similar manner.
Alina Ostafe (UNSW)
Chapter 5: The Mean Value
Senpster 1 2018
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