Finding Extrema in Functions

The video tutorial explains the concept of extrema, which includes absolute and relative maximums and minimums on a function or interval. It covers the types of extrema, the extreme value theorem, and the importance of critical numbers. The tutorial provides guidelines for finding extrema on a closed interval and includes a detailed example problem to illustrate the process.

What is another term for extrema?

• Maximums and minimums

• Derivatives

• Integrals

• Inflection points

What is the highest point on a function called?

• Absolute maximum

• Relative maximum

• Absolute minimum

• Relative minimum

Which type of extrema is the highest point in a specific area of a function?

• Relative maximum

• Absolute minimum

• Absolute maximum

• Relative minimum

Why might a function not have an absolute maximum?

• It is not continuous

• It is not differentiable

• It extends infinitely

• It is not defined

What does the Extreme Value Theorem state about continuous functions on a closed interval?

• They have both a minimum and a maximum

• They are always increasing

• They have no extrema

• They are always differentiable

What is a critical number?

• A point where the function is increasing

• A point where the function is continuous

• A point where the derivative is zero or undefined

• A point where the function is not defined

What is the first step in finding extrema on a closed interval?

• Draw the graph

• Calculate the integral

• Find the critical numbers

• Evaluate the endpoints

In the example problem, what is the critical number found for the function f(x) = -x^2 + 3x?

• x = 1

• x = 3

• x = 3/2

• x = 0

What is the absolute maximum value found in the example problem?

• 6

• 0

• 9/4

• 3

What are the absolute minimum values found in the example problem?

• 0

• 6

• 9/4

• 3