Maximum and Minimum Values of a Function

We are frequently interested in determining the function’s most significant and smallest values when given a function. This data is crucial for making accurate graphs. 

Determining the maximum and lowest values of a function is also valuable for solving optimization problems, including maximising profit, decreasing the amount of material needed in making an aluminium can, or determining the most significant height a rocket may achieve. We’ll look at how to utilise derivatives to obtain the most essential and most minor values for a function. Let’s discuss the maximum and minimum values of a function in a closed Interval with examples in detail. 

What is the Best Way to Tell if a Function has a Minimum or Maximum Value?

Identifying a function’s maximum or minimum value can be done in various ways, including by graphical tools, calculus, and so on. We’ll look at an example where we need to determine if a function has a maximum or minimum value.

To evaluate whether someone has a minimum or maximum value, we must differentiate the function and verify whether it has a positive or negative value in the provided domain.

Let’s take a look at how we arrived at our conclusion.

Explanation:

Let’s look at an example to help us comprehend. 

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

In the domain of all real numbers, we must look for their optima.

To do so, we must first divide the function twice.

⇒f'(x) = -6x + 4

⇒f”(x) = -6

We can see that the function’s double derivative is a constant of -6. As a result, it is negative.

As a result, this function has a maximum value.

Alternatively, if the double derivative for any function is positive, it seems to have a minimum.

As a result, we must twice differentiate the function and verify if it has a pro or con value in the provided domain to evaluate its minimum or maximum value.

How Would You Determine a Function’s Minimum and Maximum values?

A function correlates a value x in the first set and a value y in the second set. A function’s highest value is its most significant value. Therefore, the lowest value of any function appears to be the function’s most negligible value.

There are a few ways of determining the maximum or minimum value.

• Calculate the function’s derivative and afterwards set it to 0.

• Determine the differentiated equation’s roots.

• Change the root values in the second differentiated expression after twice differentiating the original function.

• If the value is negative, it is replaced with the root value maximum. Finally, substitute the value in the original expression for the function’s maximum.

• The minimum change in the value of the double derivative following root substitution is positive. Substitute the smallest value of the function for the value of the original equation.

• Determine the function’s highest derivatives and use the root’s value as the nth order derivative in the calculations if the second derivative is 0. If the answer is yes, the root will be given the best available function.

Properties of  Maximum and Minimum Values of a Function in a Closed Interval

• If f(x) is a continuous function in its domain, there must be at least one maximum and one minimum among equal values of f(x).

• The maxima and minima alternate. In other words, there is one minimum between two maxima and simultaneously vice versa.

• If f(x) approaches ∞ as x approaches a or b, and f'(x) = 0 only for one value x, i.e. c between a and b, then f(c) is the most minor and almost negligible value. If f(x) tends to -∞ as x tends to be a or b, f(c) is the most significant and extreme value.

Example of Maximum and Minimum Values of a Function in a Closed Interval

Let f(x)=-x2+4x+21

so,f’(x)=-x+2

As a result, we only evaluate x = 2 [0, 3]. The interval’s ends are also x = 0 and x = 3. We must now determine the value of f(x) for x = 0, 2, and 3. As a result, the absolute maximum value of f on [0, 3] is 25, and also the absolute minimum value of f on [0, 3] is 21.

Conclusion

Follow the procedure to obtain a function’s maximum and minimum values: To get to the essential stage, calculate the function’s first derivative, the roots of the differentiation function, to the function’s second derivative, applying the importance contained in the function’s second derivative. If the crucial point exchanged in the second derivative is positive or negative, determine the maximum/minimum value by swapping the critical points in the original function.