Kyle Vansant's Fourth Blog Post

How to Find Local and Global Minima and Maxima

A local minimum is the lowest point on a curve during a specific interval. Similarly, a local maximum is the highest point on a curve during a specific interval. The global minimum will be the lowest point on the entire curve, and the global maximum will be the highest point on the entire curve. 

Step 1- Find the derivative of your function.

Example- f(x)= (4x^3)-3x

                f '(x)= (12x^2)-3

Step 2- Graph your function and derivative function, if possible. This will help you identify potential minima and maxima, as well as providing you with a visual representation of the interval you will be searching in.

Step 3- Find the points on your derivative curve where f '(x)=0. These are your critical points.

Example- f '(x)= (12x^2)-3

                f ' (0)= -0.5, 0.5

Step 4- If you are finding local minima and maxima over a specific interval, disregard any critical points with x-values that are not within your interval. Calculate the endpoints of your original function on during the interval as well, and treat these points as critical points.

Step 5- Find the second derivative of your function. Graph it if possible, to provide a visual aid.

Example- f '(x)= (12x^2)-3

                 f ''(x)= 24x

Step 6- If the value of your second derivative equation is less than zero at a critical point or endpoint, that point is a maximum. If the value of your second derivative is greater than zero at a critical point or endpoint, that point is a minimum.

Example- At x= -0.5, f ''(x) is less than zero, meaning that f(-0.5) is a local maximum.

                At x= 0.5, f ''(x) is greater than zero, meaning that f(-0.5) is a local minimum.

This method of calculating minima and maxima is possible because at critical points, the slope is zero. Solving the derivative function for where x is equal to zero gives you the location of the critical points.