Increasing and Decreasing Functions

One of the most practical applications of calculus is to determine the best solution to a problem.

Introduction:

Imagine that the graph of a function represents a hill, and you are riding a bike on the hill from left to right. If you are riding uphill, the function is increasing. If you are riding downhill, the function is decreasing.

A function is increasing if, as we move from left to right along its graph, the y-coordinates increase in value.

 A function is decreasing if, as we move from left to right along its graph, the y-coordinates decrease in value.

Definition:

A function f is increasing on an interval (a, b) if f(x₂) > f(x₁) whenever x₂>x₁ in the interval (a, b).

A function f is increasing on an interval (a, b) if f(x₂)< f(x₁) whenever x₂>x₁ in the interval (a, b).

Test for increasing and decreasing:

Assume that f’(x) is defined for x ∈ (a, b).

• If f’(x) > 0 for x ∈ (a, b), f is increasing in (a, b).
• If f’(x) < 0 for x ∈ (a, b), f is decreasing in (a, b).
• If f’(x) = 0 for x ∈ (a, b), f is constant  in (a, b).

Critical point:

Critical numbers are numbers c in the domain of f(x) where f’(c) = 0, or f’(c) does not exist.

Critical points are points (c, f(c)) where c is a critical number.

Steps to find that the given function f(x) is increasing or decreasing:

• Locate critical points in the domain for f, and points where f is undefined to determine a set of open intervals whose endpoints are either critical points or points where f is undefined
• for each open interval, pick an x and find out the value of f’(x)
• Classify each interval as increasing or decreasing for f.

Examples:

f(x) =  x³+192x

f’(x) = -3x²+192

Put f’(x)=0 to get critical number.

-3x² + 192  = 0 x = 8 or -8

The numbers x=8 and x=-8 are boundary points that separate the domain into three intervals

(-∞-8)(-8,-8) and (8,0)

The graph of f(x) is