Finding Relative Extrema

Front

What are critical points in a function?

Back

Critical points are values of x where the derivative of the function is either zero or undefined.

Front

How do you find critical points of a function?

Back

To find critical points, take the derivative of the function, set it equal to zero, and solve for x. Also, check where the derivative does not exist.

Front

What does it mean when a function is decreasing?

Back

A function is decreasing on an interval if the derivative is negative throughout that interval.

Front

What does it mean when a function is increasing?

Back

A function is increasing on an interval if the derivative is positive throughout that interval.

Front

What is a relative maximum?

Back

A relative maximum is a point where the function value is higher than all nearby points, typically where the slope changes from positive to negative.

Front

What is a relative minimum?

Back

A relative minimum is a point where the function value is lower than all nearby points, typically where the slope changes from negative to positive.

Front

What is the first derivative test?

Back

The first derivative test is a method to determine whether a critical point is a relative maximum, 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 relative maximum or minimum.

Front

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

Back

If the derivative at a critical point is zero, it indicates a potential relative maximum or minimum.

Front

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

Back

If the derivative at a critical point is undefined, it may indicate a cusp or vertical tangent, which could also be a point of relative extremum.