00:01
In this question, we are given the graph of y equal to f of x and we are asked to determine the points when the first and second derivative are both positive and the points when the first and second derivative are both negative and the point when the first derivative is negative but the second derivative is positive.
00:21
So first we draw the graph of the first derivative and the second derivative using the graph of y equal to f of x.
00:30
See here, here i have drawn the graph of the first derivative to using the information that at this point the graph of the y equal to f of x is 0.
00:44
That is the derivative, the first derivative of the graph of y equal to f of x is 0 at these points.
00:51
That is this point as well as this point because we have a horizontal tangent.
00:58
So which means it will become zero here.
01:00
As well as it will become zero here so correspondingly i have drawn the points i have marked the points first and then i identified on what interval the graph of the y equal to f x the slope is positive as well as negative so if you consider the graph of the y equal to f of x starting from this interval to this interval the slope is negative which means you have to draw it like this the case i have drawn like this as you can see here till from this point to this interval the graph of the first derivative is negative and similarly from this interval to this interval the slope is positive so that is why i have drawn like this from this interval to this interval and after which the slope is negative so i have drawn it like this now considering the graph of the second derivative all that we have to look at is this point that is at this point the slope of the first derivative is itself 0.
02:04
So that is i say correspondingly i have marked this and made that i have marked this point.
02:10
And then to the left of this point, the graph of the first derivative, that is a slope of the first derivative, which in fact is the second derivative, is positive.
02:21
So that is why from this interval to this interval, i have drawn it as positive...