first derivative test problem

Front

What is the First Derivative Test?

Back

A method used to determine local extrema (maximums and minimums) of a function by analyzing the sign of its derivative.

Front

If the derivative changes from negative to positive at x = c, what can we conclude about f(c)?

Back

f(c) is a local minimum.

Front

If the derivative changes from positive to negative at x = c, what can we conclude about f(c)?

Back

f(c) is a local maximum.

Front

What does it mean if the derivative is zero at a point?

Back

It indicates a potential local extremum (maximum or minimum) or a point of inflection.

Front

What is a local maximum?

Back

A point where the function value is higher than all nearby points.

Front

What is a local minimum?

Back

A point where the function value is lower than all nearby points.

Front

What does it indicate if the derivative is positive on an interval?

Back

The function is increasing on that interval.

Front

What does it indicate if the derivative is negative on an interval?

Back

The function is decreasing on that interval.

Front

What is the significance of the critical points in the First Derivative Test?

Back

Critical points are where the derivative is zero or undefined, and they are candidates for local extrema.

Front

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

Back

By checking the sign of the derivative before and after the critical point.