 Hello students welcome to Centrum Academy channel. So today I have got for you a J advanced level question a beautiful piece of question and mind you on the exterior it might look like a very scary problem but by the time we are done with the solution I'm sure you would find this problem a really really easy problem at heart. So let's read this problem first. So it's a paragraph based question which says that let f of x be f1x minus twice of f2x where f1 and f2 functions have been defined for you over here and they have another function g of x which is basically utilizing your f of x and basically they will ask you three questions following up that concept. Now guys and girls so first of all what do you understand from the max min now these are the special type of functions which have been used very very frequently in the J advance exams so let us understand what are the meaning of min max function. So let's look at f1x so f1x talks about min of x square and mod x. So f1x talks about min of x square and mod x when x is between minus one to one and x square when mod x is greater than one. So what do you understand from the min function it is very simple guys and girls it just says that f1x what does it do it takes the minimum of the two values which is x square and mod x for a given value of x. For example if I put x value as half so it'll compare whether half square is more or more of half is more obviously more of half is more so x square which is half square will be assigned to f1x for x equal to half. So how do we actually know this function easily without having to put the values of x so here what do we do is the steps are quite simple we basically make the graphs of both these functions on the same x y coordinate system so what I'm going to do I'm going to make the graph of x square which is a parabola and mod x which is a v-shaped curve okay just between minus one to one so I'm just making this graph between minus one to one so this part is minus one this part is one okay because my sub domain here is between minus one to one. Now min means which of the two graphs is the lowest so basically you have to take that part of the graph which is the lower of the two graphs so you can see here clearly x square wins that so x square is the lower of x square and mod x so can I say now that f1x can actually be defined as x square when x lies between minus one to one and of course again x square when x mod x is greater than one here all mod x is greater than one means x greater than one or lesser than minus one so doesn't it cover everything that means f1x is going to be just an x square function for all real values of x right so f1x was quite simple in the same way we can all try out to find f2x so f2x is mod x when mod x is greater than one and max of x square mod x when mod x is less than equal to one so dear students I've already done the hard work by making the graph of mod x and x square on the same coordinate axes so here between minus one to one so when I say mod x is less than equal to one that means I'm still between minus one to one I have to take that graph which is the higher of the two so max means higher of the two so higher of the two is clearly mod x so may I say that f2x is just a mod x function because when mod x is greater than one also it follows mod x and this also gives you mod x so overall your function is just a mod x for all x belonging to real number okay so having found out my f1 and f2 it's time to find out my f of x so f of x is f1 minus 2f2x so can I say so that f1x is equal to x square minus 2 mod x okay now dear all we have been given yet another function g of x and g of x talks about the min max of f of t t lying between minus 3 to x and x itself varies between minus 3 to 0 and it takes max of the f of t if your t lies between 0 to x and x lies between 0 to 3 now this is the most scary part of this problem but I'll show you that I'll make it very easy for you to understand so for making g of x the first step is let's see how f of x behaves graphically so let me draw the graph of f of x function so I'm sure most of you have already dealt with mod functions before and this guy x square minus 2 mod x it behaves as x square plus 2x when x is less than 0 and it behaves as x square minus 2x when x is greater than equal to 0 so in this two situations let me draw the graph for x less than 0 your graph is going to look like a parabola which is going to stop here yeah getting the x axis at minus 2 and for x greater than equal to 0 it's again an upward opening parabola which will go like this cutting the x axis at plus 2 okay so this structure looks like a w to me all right now coming to my g of x so let me write my g of x once again okay so I've written the first part of the function because I want you to understand this part before we move on to the next part of the function so here our area of analysis is only between minus 3 to 0 okay so let's read the right most interval which is talking about the restriction on x value so x value is between minus 3 to 0 minus 3 included now if you take any x value in this interval then your t varies between minus 3 to that value of x so while you are taking that t which is varying between minus 3 to x you have to choose the minimum value that f of t attains okay here I want everybody to pay attention because this is very very critical if you see this point is going to be minus 1 so please look at the graph of f of x so f of x graph basically is a decreasing function from minus infinity to minus 1 right so can I say that till you reach minus 1 your function is always on the on the fall yes or no so can I say g of x is going to be is going to be your f of x value because let's say if I choose the x value right here right here then this guy has the minimum value till from minus 3 up till that point isn't it because minus 3 is higher and by the time it reaches x the value is falling down right so the function achieves its least value at that x that you have chosen at that point of time and this will happen all the way till you reach minus 1 so any x you take between minus 3 to minus 1 the value of the function at that value of x is the least of all the values that you will see of the function from minus 3 to minus 1 so can I say the function x square plus 2x itself would be the value of g of x till you reach minus 1 right now between minus 1 to 0 so I have already taken care of minus 3 to minus 1 now I have to take care of minus 1 to 0 okay I purposely included minus 1 at both the places let's see what happens now at minus 1 the least value that the function takes is minus 1 right so from minus 1 to 0 don't you realize that the least value of the function will stay at minus 1 so can I say g of x will remain minus 1 when your x starts from minus 1 and goes all the way to 0 and it was my right decision to include minus 1 at both the places because at minus 1 you get a minus 1 from this guy also and from the second guy also right so you can include minus 1 at both the places without any error introduced in the function so this defines the first part of your g of x can we similarly address the second part of g of x so second part of g of x talks about the max of f of t when t lies between 0 to x and x was chosen from 0 to 3 so let's similarly try to address this as well okay let's again look at the graph for this now here your analysis starts from 0 and all the way till 3 so do you realize that the function drops from 0 all the way till you reach 1 isn't it so the maximum value of the function will be that which it has at 0 yes the value of the function at 0 is the max till you drop to minus 1 and not only that till you go to 2 as well right so when you go from 0 to minus 1 the max value is 0 and you go from my 1 to 2 also the max value still remains the one which is at 0 or at 2 and both of them will give you 0 my dear right so here can I say that the function is as good as a 0 till you reach from 0 to 2 so the max value of the function is 0 till you move from 0 to 2 now what happens from 2 onwards from 2 onwards whatever x you take between 2 to 3 at that particular instant the function will have the maximum value isn't it so if I take a x over here so at this instant the function will have the highest value from 2 to that point x so can I say that the function will basically follow the parabolic graph when you reach from 2 to 3 okay and 2 can actually be included at both the places because at 2 both of them are going to give you a 0 so dear students after this analysis we have finally finally got our g of x so now let's go to our question the question first asks you what is the range of f of x so the range of f of x can easily be figured out from the graph of f of x so the range is from minus 1 up till plus infinity so that answers the first part of the question that is option number d which is minus 1 to infinity so let's move on to the second question the first one is already done so let's move on to the second one the number of values of x for which g of x fails to be differentiable so for this I will always recommend making the graph of g of x now this is not a very difficult graph to plot it can easily be done so let's plot the graph so g of x follows x square plus 2x when your x is between minus 3 to minus 1 so minus 3 to minus 1 let's say minus 1 is here your function follows the graph of a parabola like this right at at minus 1 it basically stops at minus 1 now from there onwards it follows the graph of a constant function which is minus 1 till you reach a 0 okay and mind you at 0 there is a hole because 0 is not included in your function now from 0 to 2 the function is going to be a flat line on the x axis so from 0 to 2 it's going to be a flat line on the x axis so from 2 to 3 it's again going to follow the graph of a parabola opening upwards now where does the function fail to be differentiable of course we know the function is going to fail to be differentiable if it shows a discontinuity at some point or it shows a corner or a cusp or a kink at some point so here i can see there's a kind of a corner getting formed here i can see there's a kind of a discontinuity and again here we see a kind of a corner getting formed so altogether there are three points of non differentiability of the function so here it answers the second part of the question so second part of the question there are three points where the function is non-differentiable so this answers the second question as well now i'm leaving it to my viewers to answer the third part of the question so please do comment in the comment section what do you think is the answer for the third part of the question so the third question says the limit of f of g of x when x tends to 0 is equal to which of the options 0 1 minus 1 are non-existent so looking forward to your comment thank you so much for watching please do like subscribe and share