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The function f(x)2x - x2 if x > 0 if x < 0(A) is discontinuous at the origin (B) is differentiable, with continuous derivative at the origin (C) is differenti...

Question

The function f(x)2x - x2 if x > 0 if x < 0(A) is discontinuous at the origin (B) is differentiable, with continuous derivative at the origin (C) is differentiable; with discontinuous derivative, at the origin (D) has a corner point at the origin (E) has a cusp at the origin

The function f(x) 2x - x2 if x > 0 if x < 0 (A) is discontinuous at the origin (B) is differentiable, with continuous derivative at the origin (C) is differentiable; with discontinuous derivative, at the origin (D) has a corner point at the origin (E) has a cusp at the origin



Answers

$$\text { Set } f(x)=\left\{\begin{array}{ll} x^{2}-x, & x \leq 2 \\ 2 x-2, & x>2 \end{array}\right.$$ (a) Is $f$ differentiable at $0 ?$ (b) Sketch the graph of $f$

You have our number 206 questions. The function affects equal to minimum 1 -1. It's a function is affect equal to minimum one minus X, comma two and X belongs to all the lumber continuous and different stable. So these are all these things need to find out. So for the best method is to graft these two functions. So this is uh let us suppose this to be y equal to one minus X and Y equals two. So first of all to draw the graph, we have to get the points of intersections. So one minus X equal to two, X equal to minus one. So when it's equal to minus one uh Why will be equal to two point? There is only one point of intersection minus one, comma two. Now let us draw the graph first days why I called to 1 -1 which is a straight line. And to draw this when X equal to zero why equal to one and when X equal to one why equal to zero? These are the two points and another second is what I call to straight line with a parallel track. Success only destroyed X. Why? When one minus one and minus? When this is too Point of intersection is -102. So this is okay -1. So why equal to 2? If you draw this line why equal to two? So let us draw like this. What I call the to line And x equal to equal to one -X. So it will pass through uh zero comma one and one comma zero. So this is zero comma one And this is 1:00. You should pass through this fine. Okay. It should first like this and this line is Why call to 1 -6. No At this point is -102. No question says minimum of this millions of this means the graph minimum means the graph should be below of the other graph. Okay, If we consider points from minus infinity to -1 Graph of Y. called to is below the graph of y equal to 1 -6. So this is minimum. And after that this graph work or to becomes above the why can't a woman sex graph line? So this will be the graph from right from here to here is the graph. We consider it at this 0.-1 comma Uh two. There is a corners where it is not different table but continues everywhere. So it is continuous everywhere but not different. She able continues everywhere but not differentiate able and X equal to -1. The answer should be continues river but not if the answer should be a Option is the correct one.

You have our number 206 questions. The function affects equal to minimum one minus X comma two. It's a function is affect equal to minimum one minus X, comma two and X belongs to all the lumber continuous and different stable. So these are all these things need to find out. So for the best method is to graft these two functions. So this is uh let us suppose this to be y equal to one minus X and Y equals two. So first of all to draw the graph, we have to get the points of intersections. So one minus X equal to two, X equal to minus one. So when it's equal to minus one uh Why will be equal to two point? There is only one point of intersection minus one, comma two. Now let us draw the graph first days why I called to one minus X which is a straight line. And to draw this when X equal to zero why equal to one and when X equal to one why equal to zero? These are the two points and another second is what I call to straight line with a parallel track. Success only destroyed X. Why? When one minus one and minus? When this is too point of intersection is minus one comma two. So this is okay minus one. So why equal to two? If you draw this line why equal to two? So let us draw like this. What I call the to line and X equal to equal to one minus X. So it will pass through uh zero comma one and one comma zero. So this is zero comma one and this is one comma zero. You should pass through this fine. Okay. It should first like this and this line is why call to one minus six. No at this point is minus one comma two. No question says minimum of this millions of this means the graph minimum means the graph should be below of the other graph. Okay, If we consider points from minus infinity to minus one, graph of Y equals two is below the graph of Y equals one million sex. So this is minimum and after that and this graph work or to becomes above the why can't a woman sex graph line? So this will be the graph from right from here to here is the graph. We consider it at this point minus one comma uh two. There is a corners where it is not different table but continues everywhere. So it is continuous everywhere but not differentiable, continues everywhere but not differentiate able and X equal to minus one. The answer should be continues river but not If the answer should be a option is the correct one.

We have the number one of the 15. Okay question says that F X equal to maximum max of one minus x and x. X squared minus one. Then F is continuous different Chevelle, that is all. So we have a fax equal to one minus x and x X squared minus one max one minus x And Access Square -1. Access Choir -1. So graphical matter is the best matter here. So to draw the graph we have to first Find the point of intersection of these two. So let us assume why equal to one minus X and y equal to access square minus one. Are the two functions So To get the point of intersection that we will be doing will be quitting tonight and said one man said An Excess Square -1. This is simply X. So access square Plus X -2 equal to zero. So this is X square -2. So let us use quality formula -7 plus minus being squired Plus four is 88 by two. X is minus one plus minus three by two, So 3 -1 is two and -2. Now we have points of intersections X coordinate of point of intersection minus two and one. So why coordinate will be equal to uh if you plug in anywhere two square four minus 13 and zero? Because one of minus one is zero point of intersections ar minus two, comma three and one comma zero. No, we can easily see that this is a straight line and this is parabola vertex vertex Equal to 0:00 -1. So let us draw both of these graph X. Why for this line if X equal to zero, X equal to zero wise one when X equal to 01 is one. So it must pass through at this point zero comma one. And again, if y is zero, Why is it lacks is one. So 1:00. So they should be there straight line. So let us draw as daughter line because we'll be doing afterwards When I'm drawing the daughter line. So this is why equal to 1 -1. And for the Parabola we have found out the vertex to be Uh the local -1. So this is minus one. So zero comma minus one is the vertex and point of intersection with the lion is minus two comma three and one comma zero. This is 1:00. So this will be minus one, This is -2. So point of intersection with the line will be somewhere here. So a parabola would look like this. Let us draw with daughter line itself. Okay, it must pass through minus one comma zero as well because Parabola is symmetrical about Y axis in this case too. Okay, no come to the now will be coming to the question point of intersection. This point of intersection is minus two comma three. No question says we have to take maximum Of 1- Sex and access par minus or wherever it is possible. So in this case maximum is for the parabola and then from the interval This -2-1, I'm talking about this -2 and this one this part that the line is above the parabola and again this is yeah, so we have this graph according to oh statement that is questions so we can easily observe that here there is a corner like this here also there is a corner so it is continuous. So our effects is continuous everywhere but not differentiable, not differentiable at minus two and X equal to one, Not differential at -2 and one but continuous everywhere. So she is the correct choice. Mhm continuous defensive. Yes she is the correct yes. Thank you so much

We have the number one of the 15. Okay question says that F X equal to maximum max of one minus x and x. X squared minus one. Then F is continuous different Chevelle, that is all. So we have a fax equal to one minus x and x X squared minus one max one minus x And Access Square -1. Access Choir -1. So graphical matter is the best matter here. So to draw the graph we have to first Find the point of intersection of these two. So let us assume why equal to one minus X and y equal to access square minus one. Are the two functions So To get the point of intersection that we will be doing will be quitting tonight and said one man said An Excess Square -1. This is simply X. So access square Plus X -2 equal to zero. So this is X square -2. So let us use quality formula -7 plus minus being squired Plus four is 88 by two. X is minus one plus minus three by two, So 3 -1 is two and -2. Now we have points of intersections X coordinate of point of intersection minus two and one. So why coordinate will be equal to uh if you plug in anywhere two square four minus 13 and zero? Because one of minus one is zero point of intersections ar minus two, comma three and one comma zero. No, we can easily see that this is a straight line and this is parabola vertex vertex Equal to 0:00 -1. So let us draw both of these graph X. Why for this line if X equal to zero, X equal to zero wise one when X equal to 01 is one. So it must pass through at this point zero comma one. And again, if y is zero, Why is it lacks is one. So 1:00. So they should be there straight line. So let us draw as daughter line because we'll be doing afterwards When I'm drawing the daughter line. So this is why equal to 1 -1. And for the Parabola we have found out the vertex to be Uh the local -1. So this is minus one. So zero comma minus one is the vertex and point of intersection with the lion is minus two comma three and one comma zero. This is 1:00. So this will be minus one, This is -2. So point of intersection with the line will be somewhere here. So a parabola would look like this. Let us draw with daughter line itself. Okay, it must pass through minus one comma zero as well because Parabola is symmetrical about Y axis in this case too. Okay, no come to the now will be coming to the question point of intersection. This point of intersection is minus two comma three. No question says we have to take maximum Of 1- Sex and access par minus or wherever it is possible. So in this case maximum is for the parabola and then from the interval This -2-1, I'm talking about this -2 and this one this part that the line is above the parabola and again this is yeah, so we have this graph according to oh statement that is questions so we can easily observe that here there is a corner like this here also there is a corner so it is continuous. So our effects is continuous everywhere but not differentiable, not differentiable at minus two and X equal to one, Not differential at -2 and one but continuous everywhere. So she is the correct choice. Mhm continuous defensive. Yes she is the correct yes. Thank you so much


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