Maxima and Minima
Finding relative maxima and minima of a function can be done by looking at a graph of the function. A relative maximum is a point that is higher than the points directly beside it on both sides, and a relative minimum is a point that is lower than the points directly beside it on both sides. Relative maxima and minima are important points in curve sketching, and they can be found by either the first or the second derivative test.
I want to talk about two new concepts relative maximum and relative minimum. Here are the definitions, a relative maximum and is sometimes called the local maximum, f has a relative maximum at x=c if of c is the largest value of f near c, and relative minimum f has a relative minimum at x=c if f of c is the smallest value of f near c. Now important for you to know because you might hear me use these terms in the videos. Plural of maximum is maxima, plural of minimum is minima.
Let's take a look at an example, the problem says identify the relative max and min of f on the interval from 1 to 15 now that's the part I got graphed here so this is my graph of y equals f of x and I want to identify where the relative maxima and relative minima occur. Well there's a bunch of them, now remember that a relative maximum if the value f of c which is larger than any value near by so, here this value 42 is a relative maximum for the function f right if you look in this little neighborhood here this is the highest point that makes it a relative maximum. So there's a relative maximum at x=1 and we can similarly go through and look for the other relative maxima there's one right here right you just cover the rest of the graph with your hands and notice that in this little interval here 35 is the highest point right 35 is the highest value that the function takes on and so there's a relative maximum at x=9.
And finally at this end point the graph seems to hit another maximum if you look at it 52 is the value of the function here and that's the highest value in some interval around that point. And so x=15 we got a relative maximum so we have 1, 9 and 15 are the locations of the relative maxima and what about the relative minima? Well it looks like these 2 and I want you to pay close attention to the fact that a lot these first of all 2 relative maxima occur at end points. End points are important to check, but also relative maxima and minima don't need to look like, they don't need to look curvy you can actually have a point in minimum or maximum so this counts as a minimum. It is the lowest point in the neighborhood around it. That value 18 is going to be the lowest point and the minimum actually occurs at x=11 and here we have a minimum value of 11 and that minimum occurs at x=5. So their relative minima at x=5 and x=11 these 2 points.
Now also important for you to know a new term relative maxima and relative minima are collectively called relative extremas. So if I say find the relative extrema I mean find the relative max and min.