0:00
Interesting question.
00:01
We are given the graph of the derivative of function, if prime x.
00:06
And we need to answer the following questions.
00:08
On what interval is fx increasing or decreasing? so pretty much straightforward if we have the graph of f prime, because it is if is increasing when when f prime x is positive, f prime x is positive.
00:24
So if prime x is positive clearly in this particular region, i'm just going to shade it.
00:30
Positive here and it's positive here.
00:34
So the answer is going to be, answer is going to be x is an element of negative 2 to 0, union 4 to infinity.
00:43
And fx is negative, sorry, fx is decreasing when f prime x is negative and clearly f prime x is negative for the remaining interval.
00:52
So x is an element of negative infinity to negative 2.
00:57
I'm still going to highlight this.
00:58
The one interval this is the second interval third interval and that's it so third interval so it's going to look like union zero to two union two two four okay for part b for what values if the x just fx has a local maximum or minima so local minima is when local minima is when f prime x changes from negative to positive changes from negative to positive and clearly it is let me just clear this up and it is changing from negative to positive and over here and over here these are the two values where it is changing from negative to positive and this is the value it is where it is changing from positive or negative here it is changing from it's not changing it's actually negative to negative so that's neither local maximum or local minima so that would imply that the local minima will be at minus 2 and 4 and likewise local maxima like i said is from f prime x from positive to negative so that is at x equal to 0 then part c is where f has inflection points so f has inflection points where f double prime x is 0 and f double prime x means 0 means f prime x is a horizontal tangent f prime x as a horizontal tangent horizontal tangent why f prime x is a horizontal tangent because f prime x would mean that this is the curve of f prime x and a tangent to f prime x will be f double prime and if the tangent is horizontal definitely the slope is going to be zero so that is a point of maxima or minima of f prime x so clearly this is a point where the tangent is horizontal this is another point this is another point this is another point and this is another point.
03:04
So all these are the points where f double prime x is zero and hence they are the point of inflection.
03:10
So it's minus one, one, two, three and five.
03:15
These are the points where f double prime x is zero.
03:18
We have to sketch the graph of a double prime x as well as a possible graph of f.
03:24
So f double prime x remember let me just clear this everything.
03:28
F double prime x is will denote the derivative of f prime x so the f prime x is increasing in this particular interval so this means that f double prime x should be positive because f double prime x is positive in the interval where f prime x is increasing and f prime x is maxima which means that f double prime x should be 0, which we already figured that these are the points where f double prime x should be zero.
04:01
So part b i'm doing over here and carefully let's try to make the graph.
04:07
First of, let's mark the points where f double prime x is 0.
04:10
So this is one of the point.
04:12
It's minus 1, then it's 1, then it's 2, then it's 3, then it's 5.
04:18
So it is positive for negative 2, for everything which is less than minus 1 so it is like this it's positive then we have from negative 1 to 1 that is decreasing so that is negative so that's going to be somewhat like this somewhat like this i'm not sure whether that can be a straight line or not so we can perhaps make a curve as well like this it can be like this and then it's going to be over at 1 it has to be 0 so it has to be to go like this then at from 1 to 2 it is increasing so f prime f double prime x is positive so it will go like this then again it is decreasing uh so that has to go like this and after 3 to 5 it is again increasing so it has to go like this then so that can be the possible curve of f double prime x and then we need to graph the possible curve of fx so remember fx is going to be dependent on f prime x because uh for the for the same reason that fx is increasing decreasing maximum minimum we have already covered that in part a to or d so at a we see that it is increasing for negative two to zero and four to infinity so let me mark the points first of that's important over here the the points are negative 2 0 2 and 4 negative 2 0 2 and 4 now local minima is at negative 2 and 4 and maxima is at 0 so negative 2 and 4 is minima and 0 is maximum and uh and i think uh that's it its inflection points are also given as minus 1, 1, 2, 3, and 5...