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 we are given, i draw the graph of f prime below, and we are being asked about the different things about the function.

00:10 So for a, we are asked about what are the interval increasing or decreasing? so for this, we simply look at when the function is positive.

00:22 So that occurs when the function is above the graph, because remember, this is f prime.

00:29 And this occurs from 0 to 2 and from 4 to 6 and then this is actually 8 right here so this is 8 sorry about that and this is from 8 to infinity so this is increasing and then it is decreasing everywhere else so that would be from 2 to 4 and 6 to 8 so just decrease it.

01:13 For local max and min, we have local backs and min.

01:19 Okay, so local max occurs when it goes from positive to negative.

01:22 So this occurs from positive to negative, so x equals 2.

01:26 So local max at 2.

01:32 And then it occurs again at 6 because it's going from positive to negative at 6.

01:39 So 2 and 6.

01:40 And then local men a curve when it goes from negative to positive.

01:48 So this is from negative to positive, so that's four, and negative to positive again at 8.

01:56 So this is 8.

01:59 For c, on one intervals, is a concave up.

02:05 So it is concave up when the slope is, so when the functions increase, it's increasing.

02:12 So conge up occurs from let's see when the slope is increasing.