What is the difference between local max and absolute max ?
sketch the graph.
a. the domain (-4,6)
b. absolute maximum of 6 at x=0
c. local maximum of 0 at x=-2
d. no absolute minimum
e. it is CU on the interval (0,6)
The domain is (-4,6); so we don't want to draw anything outside that interval, and we don't want any "holes" in that interval. (every point in that interval needs a corresponding point in the graph)
absolute max at (0,6)
So we want to keep the graph inside the dotted lines, and go through that point (0,6)
local max at (-2,0) means all the points "near -2" are less than 0, so we need something like this in the graph:
we want to have no absolute minimum. The way to do this is to have the graph head down to `- oo` . That way the points get smaller and smaller, but none of them is the smallest, because none of them is actually `-oo` .
Note: our graph doesn't actually have to be continuous, it is just that every point in the horizontal interval (-4,6) needs to have a corresponding y value (dot on the graph).
(by the way, the function I used to get that was `-((x+2)^2)/(x+4)` )
Now, we need to make it concave up from (0,6), but we need to be decreasing (or else 0 won't be an absolute max):
Note: there is an open dot at 6, because we want the domain to be (-4,6) and not (-4,6] (that is, we don't want 6 to have a spot on the graph, just everything up to 6.
Also: the function I used for the second part of the graph was: `-(x-6)^2/6` ` `
Finally, the difference between an absolute maximum and a local maximum:
an absolute max is greater than or equal to everything else. That point at (0,6) is the highest thing on the graph. a local max might also be an absolute max, but it doesn't have to be. It just has to be larger than the stuff nearby. 0 is not the tallest point on the graph. But every x value "near" -2, is less than 0.