So far we have only talked about local extrema (also called relative extrema). Local extrema are local maximums and minimums. These are also known as the highest or lowest points on the graph in that neighborhood.
Intuitive definition of absolute extrema:
- An absolute maximum is the highest y value that a function achieves.
- An absolute minimum is the lowest y value that a function achieves.
Precise definitions of absolute extrema:
- A function f has an absolute maximum at x=c, if f(c) is greater to or equal to f(x) for all x values in the domain of f. (The absolute maximum is f(c).
- A function f has an absolute minimum at x=c if f(c) is less than or equal to f(x) fo all x values in the domain of f. (The absolute minimum is f(c).
An absolute extrema will always be located at either a local extrema or endpoints of the interval. Since local extrema occur at critical values, we can also say absolute extrema occur at either critical values or endpoints of the interval.
Extreme Value Theorem
If f is continuous on the interval [a,b], then there will be absolute extrema that exist on that interval. If f is not continuous on the interval (a,b), then there could possibly be no extrema in the interval, if there is a extrema that exists it will not be an endpoint because the endpoints in the interval are not included. ( You could technically get closer and closer to endpoints of an interval)
Now that we know all about what occurs at extremas and endpoints, we can move onto optimization in calculus.
Optimization in calculus is the process of finding the greatest or least value of a function for some constraint, which must be true regardless of the solution. This part of calculus can me very confusing for some students because you have to read problems carefully to find what they are optimizing and understand what the constraints of the problem are.
When approaching an optimization problem, students need to read the problem carefully and throughly. Setting up the problem is the HARDEST part. When reading the problem, write down what the problem is asking for. Also, write down any constraints of the problem. The constraints are also known as the intervals given in the problem. For example, x values can not be less than zero.
Next you need to find the function of the problem using the constraint . For example, if we are trying to find maximum area of rectangle we would use the function: Length x Width. Make sure to add the constraint to the function so that the values that are found are true values to the problem.
Third step is to find the derivative of the function which contains the constraint. This will give you true values to the function. Once you find your true values, plug them into the ORGIONAL function ( without constraint). Note: most fo the time you have to use the constraint to find pairs of numbers, for example length and width. Make sure to test endpoints of the interval (only if included) and all true values (critical values). These numbers will be the solution to where your maximum or minimum optimization is occurring.
Take your time and read these problem slowly. They can be tricky!