Ask AI: Suppose f is a function with f(a)=b,define the following,local maximum,local minimum,maximun,minimum.

Sure, let's dive into understanding these definitions in the context of a function. 1. Local Maximum: A point in the domain of a function where the function has a value that is greater than or equal to the values of the function at all nearby points is a local maximum. It is like a "peak" in the function. 2. Local Minimum: A point in the domain of a function where the function has a value that is less than or equal to the values of the function at all nearby points is a local minimum. It's the opposite of the local maximum and resembles a "valley" in the function. 3. Maximum (also known as Absolute or Global Maximum): This is the point where the function reaches the highest value in the entire domain. If such a point exists, then it would be equal to or greater than the value of the function at every other point in its domain. 4. Minimum (also known as Absolute or Global Minimum): This is the lowest value that a function can reach within its entire domain. If such a point exists, then it would be equal to or less than the value of the function at every other point in its domain. Please note, a function might not have a global maximum or minimum. However, if the function is continuous on a closed interval [a, b], then by the Extreme Value Theorem it must attain a maximum (at some point c) and a minimum (at some point d) within that interval.