## Application of Derivatives

# Increasing and Decreasing Functions

- A function ƒ is said to be

(a) increasing on an interval (a, b) if

x_{1}< x_{2}in (a, b) ⇒ f(x_{1}) ≤ f(x_{2}) for all x_{1}, x_{2}∈ (a, b).

Alternatively, if f'(x) ≥ 0 for each*x*in (a, b)

(b) decreasing on (a, b) if

x_{1}< x_{2}in (a, b) ⇒ f(x_{1}) ≥ f(x_{2}) for all x_{1}, x_{2}∈ (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) |

+ | + | + |

+ | − | − |

− | + | − |

− | − | + |