Please explain The Extreme Value Theorem. Specifically explain how a jump discontinuity and an infinite discontinuity will prevent a maximum/minimum in their own unique way.
Assuming the function is continuous, describe the shape of potential extrema where the derivative is undefined.
Also, for a continuous function, describe the shape where the derivative is undefined.
In what scenario will a relative extrema, located at a critical number, not be an absolute extrema on an interval?
The Extreme Value Theorem (EVT) can be used to determine the presence of a minimum/maximum on a CONTINUOUS INTERVAL (very important to note). To answer your first question, any sort of discontinuity will prevent a min/max because the derivative of a point of a graph that does not exist, DOES NOT EXIST as well. Secondly, if the derivative of the function does not exist, but the function is continuous, the graph of the original equation most likely has a sharp corner where the derivative DNE, such as the function y = |x|.
To answer your final question, the relative extrema would not be an absolute extrema on an interval if, when you perform EVT and are at the point where you must test the x values, the endpoints of the interval give a value that are smaller/bigger than the where the derivative of the function equals 0. Hopefullly this helps clarify your confusion.