The graph of the derivative f' of a continuous function f is...

The graph of the derivative $ f' $ of a continuous function $ f $ is shown. (a) On what intervals is $ f $ increasing? Decreasing? (b) At what values of $ x $ does $ f $ have a local maximum? Local minimum? (c) On what intervals is $ f $ concave upward? Concave downward? (d) State the $ x $-coordinate(s) of the point(s) of inflection. (e) Assuming that $ f(0) = 0 $, sketch a graph of $ f $.

The graph of the derivative $ f' $ of a continuous function $ f $ is shown.
(a) On what intervals is $ f $ increasing? Decreasing?
(b) At what values of $ x $ does $ f $ have a local maximum? Local minimum?
(c) On what intervals is $ f $ concave upward? Concave downward?
(d) State the $ x $-coordinate(s) of the point(s) of inflection.
(e) Assuming that $ f(0) = 0 $, sketch a graph of $ f $.

Transcript

00:01 Okay, so this is the graph of f prime right here and we're being asked about different things about the function f.

00:07 So let's just go ahead and jump right into it.

00:09 So we're being asked where f is increasing or decreasing.

00:12 So we just simply look at where the function is above the x -axis for positive.

00:16 So that is increasing, but that occurs from 1 to 6, 1 to 6, and 8 to infinity.

00:30 And then it is decreasing where it is below the x -axis.

00:33 So that's from the x -axis.

00:33 0 to 1 and then from 6 to 8 oh that looks like an infinity and then for b we're being asked the local max and min so local max and local mid so in my meter local max occurs when it goes from positive to negative and then from negative and local minute it goes from negative to positive so this is going from negative to positive so that's a local mid at 1 and it occurs at 8 again.

01:15 And then a local max occurs at 6 because it's going from positive to negative.

01:23 And then concave up occurs when the slope is positive or increasing.

01:28 So this occurs from 0 to 2 and 3 to 5 and from 7 to infinity.

01:51 And then concave down occurs when the slope is negative.

01:55 And this occurs between 2 and 3 and 5 and 7.

02:04 So inflection point occurs when the slope changes, and the slope changes only when there is some sort of local max remain occurring.

02:14 So essentially where the tangent line is zero.

02:16 So if you look at the point where the tangent line is zero, you have it at two, three, five, and seven.

02:22 So those are inflection point.

02:26 So x equals two, three, five, seven.

02:33 To draw the graph of f, what we do is, i'm going to go ahead and just, draw some axis, some axes.

02:44 So we are told that it is decreasing from zero to one.

02:52 And then we have a local min at one, and that it is concave up from zero to two...

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