Applications: Maxima and Minima of Function of One Variable

APPLICATIONS: MAXIMA AND MINIMA OF FUNCTION OF ONE VARIABLE

1. Increasing and Decreasing Functions

Increasing and decreasing functions (or monotonic functions) form an important class of functions in mathematics. These functions occur in various fields

Definition 2.20

Let f be a function defined in an interval I and let x1, x2 be any two points in I

(i) f is increasing on I if f(x1) < ƒ(x2) whenever x1 < x2

(ii) f is decreasing on I if f(x1) > ƒ(x2) whenever x1 < x2

Note

1. It should be noted that the definitions of increasing and decreasing functions must be satisfied for every pair of points x1, x2 with x1 < x2.

2. To compare the function values, we have used strict in equality < (or >). So, it is sometimes referred to as ƒ is strictly increasing or strictly decreasing.

3. A function that is increasing or decreasing is said to be monotonic on the interval. The interval may be finite (i.e., bounded) or infinite (i.e., unbounded).

Theorem 2.10

Let f be a function which is continuous on the closed interval [a, b] and differentiable in the open interval (a, b)

(i) If f'(x) > 0 ∀ x ∈ (a, b), then ƒ is increasing on [a, b] 

(ii) If ƒ'(x) < 0 ∀ x ∈ (a, b), then ƒ is decreasing on [a, b] 

(iii) If ƒ'(x) = 0 ∀ x ∈ (a, b), then ƒ is constant on [a, b]

WORKED EXAMPLES

Example 1 

Find the intervals in which f is increasing or decreasing, where f(x) = 2x3 - 9x2 - 24x + 7

Solution 

The critical points divide the domain (-∞, ∞) into three non-overlapping open intervals (-∞, -1), (−1, 4), (4, ∞) on which f'(x) is either positive or negative.

Let us draw the graph of ƒ given by y = 2x3 - 9x2 - 24x + 7

When x = -1, f(−1) = −2 – 9 + 24 + 7 = 20

Example 2 

If f(x) = x3 (x − 4), then identify the intervals on which ƒ is increasing or decreasing.

Solution

The critical points divide the domain (-∞, ∞) into three non-overlapping open intervals (−∞, 0), (0, 1), (1, ∞).

The sign of f'(x) is either +ve or -ve in these intervals

Let us draw the graph of ƒ given by y = x2/3 (x - 4)

When x → 0, ƒ'(0) = -∞

⸫ y-axis is tangent at origin

Example 3 

Solution

Example 4 

Solution