global extremum

A global extremum is a minimum or maximum in the scope of a function (or within a specified open interval )

Single variable function

1. Define the function’s domain to get endpoints
2. Set all first derivatives = 0 to find critical points
3. Test whether they are local minimum or maximum using second derivative test
4. Compare the function value at the local extremum with the function value at endpoints

Let . What is the absolute minimum value of ?

Step 1 and 2: Domain & derivative

The derivative of is

• and is defined for all real numbers
• and

Step 3: Classify critical points

Evaluate the sign of on three intervals divided by two critical points 0, 2:

	-	+	+
	+	+	+
	-	+	+

At , changes it sign from - to +, so it’s a local minimum:

Step 5: Compare with endpoints

According to the table, start by going down until (0,-7) and then forever go up.

Because the function value at both endpoints are greater , 7 is the absolute minimum value.

Multivariable calculus

Linear constraints

1. Define the function’s domain  to get endpoints
2. Compute the function values at critical points
   1. Find all first partial derivatives
   2. Set partial derivatives to 0 to find critical points
   3. Only consider those that are within the domain
   4. (Optional) Using the second partial derivative test to classify critical points (in case they are saddle points)
   5. Compute the function values at critical points
3. Compute the function values at the boundaries of the domain. For each boundary
   1. Substitute it into the function to simplify the function into a single-variable function
   2. Differentiate the new single-variable function
   3. Set the derivative computed to 0 to find the value of the other variable
   4. Compute the function value
4. Compute the function values at corner points
5. Compare all values from step 2-3-4 to find absolute extremum.

Find the local and global extrema of the function such that

(Minerva Uni) (Workbook)

Step 1: Sketch the domain

Step 2: Critical points

Taking the partial derivative of :

Only the case lies inside the domain Using the second derivative test, we confirm that is a local minimum.

Step 3: Boundaries

Because we find local extrema in the domain , we must substitute the value along the boundaries

• Along the boundary , while :

• Along the boundary , while :

Step 4: Corner points

Step 5: Compare function values of local extremum, boundaries and endpoints

From previous steps, we list 5 candidates:

And corner points:

When both objective function and constraints are linear, we could use linear programming

KKT conditions

Transclude of KKT-conditions