Let’s say you are given the graph of a function, and your task is to find the maximum value. How do you proceed? This article explains how to find the maximum of a function by using the first derivative test and ending values method. If you are given a function without its graph, things get slightly trickier. You can read this article for information about how to find the maximum of a function without its graph.

**How To Find The Maximum Of A Function**

**1. Determine the First Derivative**

The first thing you need to do is test the function’s first derivative to see if the function has a local maximum or minimum at any point. If the derivative is positive at a point, the function is increasing there. If the derivative is negative, the function is decreasing there. If the local maximum has already been found, you can use the derivative to identify the point where the function begins to decrease. Likewise, if the local minimum has already been found, the derivative will tell you where the function begins to increase.

**2. Find the Local Maxima and Minima for Each Function Value**

Find all points where F’(x) < 0 and F’(x) > 0 for all x in [-π/2, π/2]. Find all points where F’(x) > 0 and F’(x) < 0 for all x in [π/2, 2π/2].

**3. Identify the Points of Maximum and Minimum for Each Function Value**

If F’(x) < 0, find the point where F’(x) = 0 and find x where F’(x) = 0. This is the value of x where the first derivative is zero. Since F’ is increasing, it must have a local maximum at this point. 2. If F’(x) > 0, find the point where F’(x) = 0 and find x where F’(x) = 0. This is the value of x where the first derivative is zero. Since F decreases, it must have a local minimum at this point and so we can conclude that this function has a local minimum there as well!

**4. Plot a Graph of Each Function Value to Find the Maximum and Minimum Points**

Plot the points (x, F’(x)) found in steps 1 and 2 on a graph of your function to find your local maximum and local minimum points. In this example, we will use a graph of y = sin x for our chart, but you can use any function you want: y = cos x, y = tan x, etc. The above graph shows all points that have been identified in step 3 above: (0,0), (π/2, π/2), (-π/2, -π/2), (π/2, -π/2 at which the function has a local maximum. that maximizes the function.

**Identifying The Points Of Inflection**

- If the function’s first derivative is positive at every point, then it has a local maximum.
- If the function’s first derivative is negative at every point, then it has a local minimum.
- If the function’s first derivative is zero at every point, then it has no local maximum or minimum. The graph of the function must be increasing or decreasing from one point to another and cannot have a local maximum or minimum anywhere in its domain.
- If the function’s first derivative is zero at a point, then it may have a local maximum or minimum near that point. However, we cannot be sure until we find the maximum or minimum itself.

**Ratio-Test For Maximum Or Minimum Value**

- Use the ratio test to find the maximum value of “f”(“x”) between “x” = 0 and “x” = 1.
- Draw a horizontal line from 0 to 1, with a slope of 1/2.
- If “f”(“x”) = 0, then there is no maximum value for “f”. This is an indication that there is a local minimum of 0. Therefore, the function does not have a local maximum or minimum at 0. Therefore, this point can be used as an ending value for the maximum search.
- If “f”(“x”) = 1 then there is no maximum value for “f”. This is an indication that there is a local minimum of 1. Therefore, the function does not have a local maximum or minimum at 1. Therefore, this point can be used as an ending value for the maximum search.
- If the derivative of “f”(“x”) is positive at the point “x” = 1, then the maximum value of “f”(“x”) is at this point.
- The first derivative test is a good method to find the maximum value of a function. However, it will not always identify an exact local maximum or minimum. It may be possible that the function has a local minimum or maximum at more than one point. In this case, you need to use another method to find the exact local maximum or minimum value of “f”.

**Using A Graph To Find The Maximum Of A Function**

- The first step is to find the maximum of the function. In other words, find the highest point on the graph.
- If a positive value has been found at some point, then a local maximum has been found. The function is increasing at that point and so the second derivative must be positive there and decreasing elsewhere. Therefore, it can be said that the function has a local maximum at that point. The corresponding value is called the maximum value or “maximum” of the function.
- If a negative value has been found at some point, then no local maximum has been found yet because we don’t know if there’s another negative value somewhere else in between where we can find another local maximum (if such exists). Therefore, one may conclude that there isn’t any other minimum in this interval for which we can say “there’s already a local minimum”. This point may be called an inflection point or an inflection value (this term was introduced by J use even before the function concept was introduced, but it is more common now). The corresponding value is called the inflection point or inflection (or sometimes a minimum) of the function.
- If a positive value has been found at some point, a negative value has been found at another point, and also no local maximum has been found, then there must be a local minimum in this interval. The first derivative must be negative at that point and increasing elsewhere. Therefore, one may conclude that there isn’t any other maximum in this interval for which we can say “there’s already a maximum”. This point may be called an asymptote or asymptotic (or sometimes a maximum) of the function. The corresponding value is called the asymptote or asymptotic maximum of the function.

**Conclusion**

The maximum of a function is the point where the function attains the greatest possible value. Finding the maximum of a function can be done in several ways, depending on whether or not you have the graph of the function. The first thing you need to do is test the function’s first derivative to see if it has a local maximum or minimum at any point. If the derivative is positive, the function is increasing locally. If the derivative is negative, the function is decreasing locally. Identifying the points of inflection is helpful when graphing the function. The ending values method is helpful when you do not have the graph. Using a graph to find the maximum of a function is helpful when you have a graph.