INCREASING AND DECREASING FUNCTIONS

To know whether a function is increasing or decreasing, we have to know where the graph of a function rises and where it falls. The graph shown below rises, falls, then rises again as we move from left to right:

It rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing when its graph rises and decreasing when its graph falls

In the above graph,

f is increasing on [a, b] and [c, d]

f is decreasing on [b, c]

Increasing and Decreasing Functions

Increasing Function :

A function f is called increasing on an interval I, if

f(x₁) ≤ f(x₂)

whenever x₁ < x₂ in I.

Decreasing Function :

A function f is called decreasing on an interval I, if

f(x₁) ≥ f(x₂)

whenever x₁ < x₂ in I.

Strictly Increasing Function :

A function f is called strctly increasing on an interval I, if

f(x₁) < f(x₂)

whenever x₁ < x₂ in I.

Decreasing Function :

A function f is called strictly decreasing on an interval I, if

f(x₁) > f(x₂)

whenever x₁ < x₂ in I.

Example 1 :

The graph shown below gives the weight W of a person at age x. Determine the intervals on which the function W is increasing and on which it is decreasing.

Solution :

The function is increasing on [0, 25] and [35, 40]. It is decreasing on [40, 50]. The function is constant (neither increasing nor decreasing) on [25, 35] and [50, 80]. This means that the person gained weight until age 25, then gained weight again between ages 35 and 40. He lost weight between ages 40 and 50.

Example 2 :

(a)  Sketch the graph of the function f(x) = x^2/3.  

(b)  Find the domain and range of the function.

(c)  Find the intervals on which f increases and decreases.

Solution :

(a)  We can use a graphing calculator to sketch the graph shown below.  

From the graph above, we can observer that

(b)  the domain of f is R and the range is [0, ∞). 

(c)  f is decreasing on (-∞, 0] and increasing on [0, ∞). 

Example 3 :

The graph of the function shown below has domain [−1, 6].

(a) Find the largest interval on which f is increasing.

(b) Find the largest interval on which f is decreasing.

(c) Find the largest interval containing 6 on which f is decreasing.

Solution :

From the graph above, we can observe,

(a) [1, 5] is the largest interval on which f is increasing.

(b) [−1, 1] is the largest interval on which f is decreasing.

(c) [5, 6] is the largest interval containing 6 on which f is decreasing.

Example 4 :

Shown below are the graphs of three functions; each function is graphed on its entire domain.

(a) Is f increasing, decreasing, or neither?

(b) Is g increasing, decreasing, or neither?

(c) Is h increasing, decreasing, or neither?

Solution :

(a) The graph of f gets lower from left to right on its entire domain. Thus f is decreasing.

(b) The graph of g gets higher from left to right on its entire domain. Thus g is increasing.

(c) The graph of h gets lower from left to right on part of its domain and gets higher from left to right on another part of its domain. Thus h is neither increasing nor decreasing.

Example 5 :

Here f has domain [0, 4] and g has domain [−1, 5].

(i)  What is the largest interval contained in the domain of f on which f is increasing?

(ii)  What is the largest interval contained in the domain of g on which g is increasing?

Solution :

(i) The largest interval on which the function f increasing is [3, 4].

(i) The largest interval on which the function g decreasing is [0, 3].

Example 6 :

Find the intervals in which f(x) = 2x³ + x² - 20x is increasing and decreasing.

Solution :

f(x) = 2x³ + x² - 20x

f'(x) = 6x² + 2x - 20

f'(x) = 2(3x² + x - 10)

f'(x) = 2(x + 2)(3x - 5)

To find critical numbers, equate f'(x) to zero and solve for x.

f'(x) = 0

2(x + 2)(3x - 5) = 0

(x + 2)(3x - 5) = 0

x + 2 = 0  or  3x - 5 = 0

x = -2  or x = ⁵⁄₃

Critical numbers are -2 and ⁵⁄₃.

The values -2 and ⁵⁄₃ divide the real line (the domain of f(x)) into intervals (-∞, -2), (-2, ⁵⁄₃) and (⁵⁄₃, ∞).

Notes :

(i) If the critical numbers are not included in the intervals, then the intervals of increasing (decreasing) becomes strictly increasing (strictly decreasing).

(ii) The intervals of increasing/decreasing can be obtained by taking and checking a sample point in the sub-interval.

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