Understanding Derivatives and Extrema

The video tutorial explains how to analyze the graph of a derivative function to determine where the original function is increasing or decreasing and to identify any relative extrema. It covers identifying critical points where the derivative is zero or undefined, determining intervals of increase and decrease, and finding relative maxima and minima. The tutorial emphasizes understanding the relationship between the derivative's sign and the behavior of the original function.

What does a positive derivative indicate about the function over an interval?

• The function is decreasing.

• The function is constant.

• The function is increasing.

• The function is undefined.

At which x-value is the derivative undefined?

• x = 10

• x = 2

• x = 6

• x = 8

Over which interval is the derivative negative?

• From 2 to 6

• From 0 to 2

• From 8 to infinity

• From 6 to 10

What is the behavior of the function over the interval from 8 to infinity?

• The function is constant.

• The function is increasing.

• The function is decreasing.

• The function is undefined.

Where does the function have a relative minimum?

• At x = 4

• At x = 10

• At x = 6

• At x = 2

What change in the derivative indicates a relative maximum?

• From positive to negative

• From negative to positive

• From zero to undefined

• From undefined to zero

What is the significance of the derivative being zero at a point?

• The function is increasing at that point.

• The function has a potential extremum at that point.

• The function is undefined at that point.

• The function has a constant slope at that point.

What does the continuity of the function imply about its derivative at x = 6?

• The function is undefined.

• The derivative is zero.

• The derivative is positive.

• The function is defined even if the derivative is not.

What happens to the function at x = 8?

• It has a relative maximum.

• It has a relative minimum.

• It is undefined.

• It is constant.

What does a change from negative to positive in the derivative indicate?

• A relative maximum

• A relative minimum

• A constant function

• An undefined function