Using the First Derivative Test in Calculus

Front

What is the First Derivative Test?

Back

A method used to determine the local maxima and minima of a function by analyzing the sign changes of its first derivative.

Front

What does it mean for a function f(x) to be increasing on an interval?

Back

f(x) is increasing on an interval if f'(x) > 0 for all x in that interval.

Front

What does it mean for a function f(x) to be decreasing on an interval?

Back

f(x) is decreasing on an interval if f'(x) < 0 for all x in that interval.

Front

What are critical points of a function?

Back

Critical points are values of x where f'(x) = 0 or where f'(x) is undefined.

Front

True or False: All critical points are extrema.

Back

False. Not all critical points are local maxima or minima.

Front

How do you determine if a critical point is a local maximum or minimum using the First Derivative Test?

Back

Check the sign of f'(x) before and after the critical point. If it changes from positive to negative, it's a local maximum; if it changes from negative to positive, it's a local minimum.

Front

What is the significance of a positive first derivative?

Back

A positive first derivative indicates that the function is increasing.

Front

What is the significance of a negative first derivative?

Back

A negative first derivative indicates that the function is decreasing.

Front

If f'(x) changes from positive to negative at x = c, what can be concluded about f(c)?

Back

f(c) is a local maximum.

Front

If f'(x) changes from negative to positive at x = c, what can be concluded about f(c)?

Back

f(c) is a local minimum.