Understanding Extreme Values of Functions

What is the absolute minimum value of the function y = x^2 over the entire real number domain?

• Infinity

• Negative Infinity

• 0

• 1

For the function y = x^2 on the domain from 0 to 3, what is the absolute maximum value?

• 0

• 6

• 9

• 3

According to the Extreme Value Theorem, what must be true for a function to have both a maximum and minimum value on an interval?

• The function must be quadratic

• The function must be linear

• The function must be continuous

• The function must be differentiable

What is a local maximum in terms of the points around it?

• It is lower than all points around it

• It is higher than all points around it

• It is the lowest point in the entire domain

• It is the highest point in the entire domain

What is a critical point in the context of derivatives?

• Where the derivative is positive

• Where the derivative is negative

• Where the derivative is zero or does not exist

• Where the derivative is constant

What is the difference between a critical point and a stationary point?

• There is no difference

• A critical point is where the derivative is zero or undefined, a stationary point is where it is zero

• A critical point is where the derivative is zero, a stationary point is where it is non-zero

• A critical point is where the derivative is non-zero, a stationary point is where it is zero

When finding extreme values on an interval, what must you consider besides critical points?

• Only the end of the interval

• Only the start of the interval

• Both the start and end of the interval

• Neither the start nor the end of the interval

In the example given, what is the absolute maximum value found for the function on the interval?

• 0.685

• 1.933

• 0.429

• 2.000

What is the absolute minimum value found for the function on the interval in the example?

• 2.000

• 0.685

• 1.933

• 0.429

Why is it important to check if a function is continuous when applying the Extreme Value Theorem?

• Because discontinuous functions are always differentiable

• Because continuous functions are always differentiable

• Because the theorem only applies to continuous functions

• Because discontinuous functions have no extrema