Increasing and Decreasing functions 2023

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What is an increasing function?

Back

A function is increasing on an interval if, for any two points x1 and x2 in that interval, where x1 < x2, the function value at x1 is less than the function value at x2 (f(x1) < f(x2)).

Front

What is a decreasing function?

Back

A function is decreasing on an interval if, for any two points x1 and x2 in that interval, where x1 < x2, the function value at x1 is greater than the function value at x2 (f(x1) > f(x2)).

Front

How do you identify intervals of increase or decrease from a graph?

Back

To identify intervals of increase or decrease from a graph, look for sections where the graph rises (increasing) or falls (decreasing) as you move from left to right.

Front

What is a local maximum?

Back

A local maximum is a point on the graph of a function where the function value is higher than the values of the function at nearby points.

Front

What is a local minimum?

Back

A local minimum is a point on the graph of a function where the function value is lower than the values of the function at nearby points.

Front

How do you find local maxima and minima using derivatives?

Back

To find local maxima and minima, take the derivative of the function, set it to zero to find critical points, and use the second derivative test to determine if each critical point is a maximum or minimum.

Front

What does it mean for a function to be constant?

Back

A function is constant on an interval if it does not change value; that is, for any two points in the interval, the function value is the same.

Front

What is the significance of the first derivative test?

Back

The first derivative test helps determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of the derivative before and after the critical point.

Front

What is the significance of the second derivative test?

Back

The second derivative test determines the concavity of the function at a critical point; if the second derivative is positive, the function is concave up (local minimum), and if negative, it is concave down (local maximum).

Front

How do you express intervals in interval notation?

Back

Intervals in interval notation are expressed using parentheses for open intervals (not including endpoints) and brackets for closed intervals (including endpoints). For example, (a, b) or [a, b].