Local Maxima and Minima

Local Maximum

For a function y=f(x), x-x₀ is a point of local maximum if there exists an open interval containing x₀ such that f(x₀)>f(x) for all values of x lying in that interval. The corresponding value of y(=y₀=f(x₀)) is known as local maximum value.

Local Maximum

For a function y=f(x), x-x₀ is a point of local maximum if there exists an open interval containing x₀ such that f(x₀) < f(x) for all values of x lying in that interval. The corresponding value of y(=y₀=f(x₀)) is known as local minimum value.

In the fig. shown;

x=a and x=c are the two points of location maxima.

x=b and x=d are the two points of location minima.

Steps to determine points of local maxima and local minima

At point of local maxima and local minima the slope of tangent drawn to the curve is zero. i.e. if x=a is either point of local maximum or local minimum of function y=f(x), then f’(a) = 0. So to obtain points of local maxima and minima, find f’(x) and equate it to zero. Then to decide between local maxima and minima we can use any one of the following methods:

Method – 1 (First Derivative Test)

• Find the value(s) of x where f’(x) vanishes.

Let x=x₀ be one of these values.

• Local Maximum

x=x₀ is a point of local maximum if:

f’(x) > 0 at every point close to and to the left of x₀ and f’(x) < 0 at every point close to and to the right of x₀.

OR

f’(x₀-h) > 0 > 0 and f’(x₀+h) > 0, where is a small +ve number

• Local Minima

x=x₀ is a point of local minimum if:

f’(x) < 0 at every point close to and to the left of x₀ and f’(x) > 0 at every point close to and to the right of x₀.

OR

f'(x₀-h) < 0 and f’(x₀+h) > 0, where is a small +ve number

Method – 2 (Second Derivative Test)

• Find the values of x where f’(x) vanishes.

Let x=x₀ be one of these values.

• Local Maximum

x=x₀ is a point of local maximum if f”(x­₀) < 0.

• Local Minima

x=x₀ is a point of local minimum if f”(x­₀) > 0.