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Numerade Educator

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Problem 16 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) = 2 - \frac{1}{3}x $, $ x \geqslant -2 $

Answer

Absolute maximum $f(-2)=\frac{8}{3}$. No local maximum. No absolute or local minimum.

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Video Transcript

let's sketch the graph of the function F of x equal to minus one third time sex For X greater than or equal to -2. And we will use that graft to find the absolute and local maximum and minimum values of the function. So we have 2 -1 3rd time sex for eggs on the interval or X in the interval from negative to to the right. That is up to plus infinity. Close at -2. I'm gonna go this line that is determined completely by two points. So uh X equals negative two. The image is two minus one third times negative two, which is two plus two thirds. And that is 33 times two is six plus two is eight thirds. So the image at -2 is 8/3. And the other the other point we can use is for X equals zero. We get to so it's zero. The images to is here. And this is the image of negative to which we draw with uh feel red circle and then the function decreases all the time to the right without any bound. So um with this behavior of the functions line in fact we can talk about a little bit before giving the solution to the problem that this graph can also be obtained by some transformation to the identity function. In fact we have the identity function. It is. Then this function is multiplied by 1/3. That makes the function tilt to the right a little bit because now the image of one is one third. So we have some kind of turn to uh the line is closer to the X axis. Now then we have negative that is a reflection with respect to the X axis, something like this. And then after that we shift really displace the graph up, achieved the graph up two units. So we get this or less and that's it. That's what we found here this line. So there's another way. But each time we have a craft which is functions a linear function. We can throw it the graphics line and we can write just determining two points. Okay so um so now if we look carefully at the graph we can see that we have an absolute maximum value Equal to 8/3 attained at -2 which is this value here which is this value here you see the highest point on the graph and it's included in the graphic of excellent attitude is included in the domain we are considering so we can stay here. That gap has Yeah it has an absolute maximum value eight thirds and that value of course At x equals -2. There is no local maximum. Okay, that's because at any point of the under graph we have images that are greater and less than the image at the point. So there's no local maximum. Yeah and in the case of the answer of maximum it is not local because we had no crafted left in fact. Okay so we respect to the minimum we had no outside the minimum because this graph is decreasing all the time to the right of the values effects that is, there is no lower bound to the graph. So I have has no absolute but also local. There is no local minimum because for any point in the draft there are points with greater images that are greater or smaller than the image of the point, so it has no absolute or local minimum. Told the only thing that this graph has is an absolute maximum value of eight thirds. Which of course at Mexico native to If we had considered this graph open on negative to that is if this point here is not included in the graph, the function we would have no external at all, but in this case is included, so it's the absolute maximum value, that's it for this function.