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you're given the graph of f ' what does f ' say about f itself ?
critical points are where f ' equals 0
and critical points are possible extrema
F ' ALSO SHOWS SLOPE
sorry caps lock was on
exactly, f ' tells us the slope of the tangent lines of f translation: we can figure out where f is increasing or decreasing based on the graph of f '
where is f increasing? where is f decreasing?
f is increasing on intervals [-2,2] amd [4,6]
careful, this isn't the graph of f
This is the graph of f ', the derivative of f. So what you need to do is look to see where f ' is negative to figure out where f is decreasing. On the flip side, look where f ' is positive to see where f is increasing |dw:1449364728050:dw| in the drawing, the shaded region represents where f ' is negative ---> f is decreasing
so x = 5 is the min
no wait, that's a local min right? not an absolute
yes because the graph of f decreases until you hit x = 5. After that x value, it starts to increase again so we have something like this for f itself |dw:1449364986531:dw| one thing to note: the only point we know on f(x) is (3,10). The actual y coordinate of the min of f could be any value really. The min point doesn't necessarily have to be below the x axis
it's both local and absolute
this region here has f decreasing |dw:1449365189587:dw|
Okay, thank you. You're good with this. :)
I have a couple more. Is that ok?
so we start up high somewhere, then we decrease (passing through (3,10)) until we bottom out at (5,y) where y is unknown then we start to increase again because of this positive region on f ' |dw:1449365251674:dw|