extrema of functions

Front

What is an extremum in the context of functions?

Back

An extremum is a point in the domain of a function where the function takes on a maximum or minimum value.

Front

How do you find the extrema of a function?

Back

To find the extrema, take the derivative of the function, set it to zero, and solve for the critical points. Then, use the second derivative test or the first derivative test to determine if these points are maxima or minima.

Front

What is a critical point?

Back

A critical point is a point in the domain of a function where the derivative is either zero or undefined.

Front

What does it mean for a function to be increasing or decreasing?

Back

A function is increasing on an interval if the output values rise as the input values increase. It is decreasing if the output values fall as the input values increase.

Front

What is the first derivative test?

Back

The first derivative test is a method used to 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 second derivative test?

Back

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

Front

What is the difference between local and global extrema?

Back

Local extrema are the highest or lowest points in a specific neighborhood of the function, while global extrema are the highest or lowest points over the entire domain of the function.

Front

How do you evaluate a function at a specific point?

Back

To evaluate a function at a specific point, substitute the value of the variable into the function and simplify.

Front

What is the significance of the y-intercept in a linear function?

Back

The y-intercept is the point where the graph of the function crosses the y-axis, indicating the value of the function when the input (x) is zero.

Front

What is a linear function?

Back

A linear function is a function that can be expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept.