Application of Derivatives

Increasing and Decreasing Functions


  • A function ƒ is said to be
    (a) increasing on an interval (a, b) if
    x1 < x2 in (a, b) ⇒ f(x1) ≤ f(x2) for all x1, x2 ∈ (a, b).
    Alternatively, if f'(x) ≥ 0 for each x in (a, b)
    (b) decreasing on (a, b) if
    x1 < x2 in (a, b) ⇒ f(x1) ≥ f(x2) for all x1, x2 ∈ (a, b).
    Alternatively, if f'(x) ≤ 0 for each x in (a, b)

Monotonic functions-Tips:

  • A function f(x) is strictly increasing on R if f'(x) > 0 ∀ x ∈ R.
  • A function f(x) is strictly decreasing on R if f'(x) < 0 ∀ x ∈ R.
  • A function f(x) is increasing on R if f'(x) ≥ 0.
  • A function f(x) is decreasing on R if f'(x) ≤ 0.

View the Topic in this video From 10:26 To 20:04

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  • A function f(x) = \frac{a\cos x + b \sin x}{c\cos x +d\sin x} is increasing if ad − bc < 0
  • If f(x) and g(x) are continuous and differentiable function and fog and gof exists, then
f'(x) g'(x) (fog)'(x) (or) (gof)'(x)
+ + +
+ − −
− + −
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