Question(1)
4. Give an example of a function with no absolute minimum or maximum.
Studdy Solution
STEP 1
1. We are looking for a function that has no absolute minimum or maximum values.
2. The function should be defined on a domain that allows for this property.
STEP 2
1. Understand the concept of absolute minimum and maximum.
2. Choose a suitable type of function.
3. Define the function.
4. Verify that the function has no absolute minimum or maximum.
STEP 3
Understand the concept of absolute minimum and maximum.
An absolute minimum of a function is the lowest point over its entire domain, and an absolute maximum is the highest point. A function with no absolute minimum or maximum does not have these extremum points over its domain.
STEP 4
Choose a suitable type of function.
A common type of function that has no absolute minimum or maximum is a linear function with a non-zero slope, or a function defined on an unbounded domain like the entire set of real numbers.
STEP 5
Define the function.
Let's choose the linear function , which is defined for all real numbers .
STEP 6
Verify that the function has no absolute minimum or maximum.
For the function , as approaches positive infinity, also approaches positive infinity, and as approaches negative infinity, approaches negative infinity. Therefore, there are no absolute minimum or maximum values.
The function is an example of a function with no absolute minimum or maximum.
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