Theorem 1 Let  f  be continuous on [a, b] and differentiable on the open interval
(a,b) Then
(a)
f  is increasing in [a,b] if f ′(x) > 0 for each x ∈ (a, b)
(b)
f  is decreasing in [a,b] if f ′(x) < 0 for each x ∈ (a, b)
(c)
f  is a constant function in [a,b] if f ′(x) = 0 for each x ∈ (a, b)
Strictly Increasing function
(i)
Neither Increasing nor
Decreasing function
(iii)
Strictly Decreasing function
(ii)
Rationalised 2023-24
 MATHEMATICS
154
Proof (a) Let x1, x2 ∈ [a, b] be such that x1 < x2 Then, by Mean Value Theorem (Theorem 8 in Chapter 5), there exists a point c
between x1 and x2 such that
f(x2) – f(x1) = f ′(c) (x2 – x1)
i e