Maxima and Minima of Functions

Maxima and minima of a function are the largest and smallest value of the function respectively either within a given range or on the entire domain. Collectively they are also known as extrema of the function. The maxima and minima are the respective plurals of maximum and minimum of a function. Before understanding maxima and minima in detail, let’s understand the local maximum and minimum value of the function first.

Local Maximum and Minimum

For finding maximum and minimum of a function, first we need to choose an interval.

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Local maximum and minimum are shown in the above graph. Let’s understand one by one in detail.

Local Maximum:

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In the above graph f(x) is a function. Interval is shown around the point a. We can say that a local maximum is a point where the height of the function at point "a" is greater than (or equal to) the height anywhere else in that interval.

Briefly, we can write 

• f(a) ≥ f(x) for all x in the interval.

In other words, we can say that there is no height greater than f(a).

Always keep in mind that a should be inside the interval, not at one end or the other.

Local Minimum:

It is similar to a local minimum, the only difference here is function value at point “a” in the interval is less than (or equal to) the height anywhere in that interval.

Briefly, we can write:

• f(a) ≤ f(x) for all x in the interval.

Global (or Absolute) Maximum and Minimum:

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• The maximum or minimum over the entire domain of the function is called an "Absolute" or "Global" maximum or minimum.

• There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum of a function.

Finding Maxima and Minima Using Derivatives

Maxima and minima are found by using the concept of derivatives. As we know the concept of derivatives gives us the information regarding the gradient or slope of the function, we locate the points where the gradient is zero and these points are called turning points or stationary points. These are points associated with the largest or smallest values (locally) of the function.

How to Calculate Maxima and Minima Points?

Let’s understand this with an example.

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From the above figure, we can see before the slope becomes zero it is negative after it gets zero and again it becomes positive. It can be said dy/dx is -ve before stationary point dy/dx is +ve after a stationary point. Hence it can be said d2y/dx2 is positive at the stationary point shown. Therefore we can say that wherever the double derivative is positive it is the point of minima. Vice versa we can also say wherever the double derivative is negative it is the point of maxima on the curve. This is also called the second derivative test.

Derivative Tests:

To find the maxima and minima of any function we use the derivative test. Generally, first-order derivative and second-order derivative tests are used. Let us have a look in detail.

First Order Derivative Test:

Consider f be the function defined in an open interval I. Also, f be continuous at critical point c in I such that f’(c) = 0.

• If f’(x) changes sign from positive to negative as x increases through point c, then c is the point of local maxima, and f(c) is the maximum value.

• If f’(x) changes sign from negative to positive as x increases through point c, then c is the point of local minima, and f(c) is the minimum value.

• If f’(x) doesn’t change sign as x increases through c, then c is neither a point of local minima nor a point of local maxima. 

Second Derivative Test:

Consider f be the function defined on an interval I and it is twice differentiable at c.

• If  x = c will be the point of local maxima if f'(c) = 0 and f”(c)<0. Then f(c) will be having local maximum value.

• If x = c will be the point of local minima if f'(c = 0 and f”(c) < 0. Then f(c) will be having local minimum value.

• When both f'(c) = 0 and f”(c) = 0 the test fails. And that first derivative test will give us the value of local maxima and minima.

Properties of maxima and minima are as follow :

• If f(x) is a continuous function in its domain, then at least one maximum or one minimum should lie between equal values of f(x).

• Maxima and minima occur alternately. i.e between two maxima there is one minima and vice versa.

• If f(x) tends to infinity as x tends to a or b and f’(x) = 0 only for one value x i.e.c between a and b, then f(c) is the minimum and the least value. If f(x) tends to – ∞ as x tends to a or b, then f(c) is the maximum and the highest value.

How to Find Maxima Functions?

To find the maxima of a function, we need to find a derivative of a function f(x) and find the critical numbers. Then, find the second derivative of a function f(x) and put the critical numbers. If the value is negative, the function has relative maxima at that point, if the value is positive, the function has relative maxima at that point.