Final answer:
When asked to find the local maximum or minimum values of a function on an interval, you need to locate the highest or lowest points within that interval, called local extrema. To find these extrema, calculate the derivative of the function, find the critical points, and evaluate the function at these points and the endpoints of the interval.
Step-by-step explanation:
Mathematics - High School Level
When you are asked to find the local maximum or minimum values of a function on a given interval, you are looking for the highest or lowest points that the function reaches within that interval. These points are called local extrema. Local maximum values are the highest points, while local minimum values are the lowest points.
To find the local extrema of a function, you need to find the critical points where the derivative of the function equals zero or is undefined. Then, you evaluate the function at these critical points as well as the endpoints of the interval to determine which points are local extrema.
For example, let's say you have the function f(x) = x^2 - 4x + 3. To find the local extrema on the interval (-2, 2), you need to:
- Calculate the derivative of f(x) to find the critical points.
- Set the derivative equal to zero and solve for x to find the critical points.
- Evaluate f(x) at the critical points and the endpoints of the interval to determine the local extrema.