Extrema on an Interval/ First Derivative Test

Front

What is a critical point of a function?

Back

A critical point of a function g(x) occurs where the derivative g'(x) is either zero or undefined. It is a point where the function may have a local maximum, local minimum, or an inflection point.

Front

What does it mean for a function to be increasing on an interval?

Back

A function f(x) is said to be increasing on an interval if for any two points a and b in that interval, where a < b, the value of f(a) < f(b).

Front

What does it mean for a function to be decreasing on an interval?

Back

A function f(x) is said to be decreasing on an interval if for any two points a and b in that interval, where a < b, the value of f(a) > f(b).

Front

What is the First Derivative Test?

Back

The First Derivative Test is a method used to determine the local maxima and minima of a function by analyzing the sign of the derivative before and after a critical point.

Front

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

Back

If g'(x) changes from positive to negative at x = c, then g(c) is a local maximum.

Front

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

Back

If g'(x) changes from negative to positive at x = c, then g(c) is a local minimum.

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes, which can be identified by a change in the sign of the second derivative.

Front

How do you find intervals where a function is increasing or decreasing?

Back

To find intervals where a function is increasing or decreasing, calculate the first derivative, set it to zero to find critical points, and test the sign of the derivative in the intervals defined by these points.

Front

What is the significance of g'(x) = 0?

Back

The equation g'(x) = 0 indicates that x is a critical point, which may correspond to a local maximum, local minimum, or an inflection point.

Front

What does it mean if a function is constant on an interval?

Back

A function is constant on an interval if it takes the same value for every point in that interval, meaning its derivative is zero throughout that interval.