Problem 25 Easy Difficulty

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 1 - \sqrt{x} $

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Fahad P.

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Chris T.

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Video Transcript

we are going to sketch the graph of the function one minus square root of x. And we'll use that graph to find the absolute and local maximum and minimum values of the function. So there is no domain specified here. So we are going to use the domain implicit in the formula that is in this case we have a square root of X. And that needs or um is only defined for X greater than or equal to zero. So we are considering F of X equals one minus squared of X. For eggs on the interval zero including 0 and plus infinity. Yeah, that is for the non negative number eggs. Then we know that the square root of X is inverse of the um of X square for the positive numbers. That is we're going to do some transformation to arrive to the graph of as for its of X. So we have this let's say the Parabola X square, something like this. It has a negative part. But we don't, that is a part which corresponds to the images of the negative numbers which is just to reflect a reflection of these respect to the Y axis. But we know that we have to consider only X positive. So this is the part of the parabola X where we got to consider. So here we have let's say Y equals X square for X greater than records. Era. And we know that to find that the verse of that we get to draw the identity function. That is the line why it was what Y equals X. Let's say that. And with that we made a reflection respect to that line and we get the inverse of the function. So we are going to get something just a little bit like this. It's a something that's not so bad like this, but trying to make a little bit idea of their reflection. Okay. And it's endless to the right, that's what's the problem. So now we take out the problem here for a moment that people already found inverse of the parabola. And we have this that's square the facts now. And how we did it. We draw the parabolic X square for the positive X. Only because if we take all the Parabola is not an objective function. So we need to take on the the positive parties for the images for the positive X of the problem, withdraw the identity line and we made reflections through the line. We know that the inverse of the functions are constructed that way. So now we have the square root and now we get to change sign of that that is negative square root of X. It and that means a reflection respect to the X axis. So we get these more or less. So this is negative spirit of X now. And we can get rid of this for the moment. Yeah. And we have this, we got to someone to add one to get F. Finally, so to get f. We at one and we know that that we had to displace the function completely as is one unit upward. That is because all the images of this function here Get to be at one. And that makes a replacement displacement of one upward. So we can say we have now this function more or less. So we can say that we can get you can get rid of the previous graph. We have already are function here. Now we have some values we recognize here because with the place one at zero we get one visible to one. So the point is included in the graph because for X equals zero. The functions well defined. And this body here responded to the intersection with the X. Axis is now uh Mexico one. Okay, so to hear X Equal one because when we put Mexico and in our formula here we get to Rome and after that the function is decreasing all the time without any bound. Yeah. And that's the idea. So we have a function which is always increasing at zero. We have value one at one. We have Alice era and the function degrees without any bound. So it is clear that there is no absolute minimum because the function it doesn't stop to decrease. So we have no lowest point in the graph. Of course no local maximum or minimum because at any point of the graph we have always values greater and smaller than the value at the point And we have as the highest point on the graph which is zero At zero which is a valuable one. So we have an absolute maximum one at zero. And it's not a local maximum because we have no graph to the left. So we can write all these observation here F has an absolute maximum value one which accor's at X equals zero. Okay. And let's say here F has no local maximum at all because at any point we have a smaller and greater valid that values than the image of that point. If we take any close any small interval contain the point. So we have this this and now for the minimum we don't have either local nor absolute minimum. Have has no yeah no absolute or minimum or local salary. Mhm minimum. And that's it. That's the behavior of the function one minus scores of X. Which we found by doing some transformation to the basic functions first we took these parabola X square for the positive values of X. Within a reflection of that we respect to the identity line, Y equals X. We did that because that's the way of graphically finding the inverse of the function after that we changed the sign of the function with was a reflection respect to the X axis and then we displaced upward one unit. And with all that those transformations we get function F which is this red line here because of the behavior of always decrease enough to function. We know that there is no local normal maximum, our local or absolute minimum, there is also no local maximum. There is there is an absolute maximum value at one at this point here, the Valley one, and that of course at the value of X equals zero. So this is the analysis for this function. Okay?