How to determine whether a function has a minimum or maximum value?

Maxima and minima are very important and interesting concepts in the field of mathematics. Finding the maximum or minimum value of a function can be done in various ways like using graphical techniques, using calculus etc. We will have a look at an example in which we have to determine whether a function has a maximum or a minima value.

Answer: To determine whether a function has a minimum or maximum value, we have to double differentiate the function and check whether it has a negative or a positive value in the given domain.

Let us understand how we arrived at the answer.

Explanation:

Let us understand this with the help of an example.

Let us assume a function f(x) = -3x² + 4x + 7

We have to check for its optima in the domain of all real numbers.

For that, firstly, we have to differentiate the function two times.

⇒f'(x) = -6x + 4

⇒f''(x) = -6

Here, we find that the double derivative of the function is a constant -6. Hence, it is negative.

Hence, this function has a maximum.

Alternately, if the double derivative comes out to be positive for any function, then it has a minimum.

Hence, to determine whether a function has a minimum or maximum value, we have to double differentiate the function and check whether it has a negative or a positive value in the given domain.