EVT & MVT Practice

Front

What is the Extreme Value Theorem (EVT)?

Back

The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then it has both a maximum and a minimum value on that interval.

Front

What is the Mean Value Theorem (MVT)?

Back

Front

What is a critical point?

Back

A critical point of a function is a point in its domain where the derivative is zero or undefined.

Front

How do you find the absolute maximum of a function on a closed interval?

Back

To find the absolute maximum, evaluate the function at the critical points and at the endpoints of the interval, then compare these values.

Front

What does it mean for a function to be differentiable?

Back

A function is differentiable at a point if it has a defined derivative at that point, meaning it has a tangent line.

Front

What is the relationship between EVT and differentiability?

Back

The Extreme Value Theorem does not require the function to be differentiable, only continuous on a closed interval.

Front

What is the significance of the point c in the Mean Value Theorem?

Back

The point c represents a point in the interval (a, b) where the instantaneous rate of change (slope of the tangent) equals the average rate of change over the interval.

Front

How do you determine if the Mean Value Theorem applies to a function?

Back

To apply the Mean Value Theorem, the function must be continuous on [a, b] and differentiable on (a, b).

Front

What is the first derivative test?

Back

The first derivative test is a method to determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of the derivative before and after the point.

Front

What is the second derivative test?

Back

The second derivative test uses the second derivative of a function to determine the concavity at a critical point, helping to classify it as a local maximum or minimum.