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Problem 23 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) = \ln x $, $ 0 < x \leqslant 2 $

Answer

Absolute maximum
$f(2)=\ln 2 \approx 0.69$. No local maximum. No absolute
or local minimum.

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Video Transcript

you're gonna sketch to graph of the function natural algorithm of X for x greater than zero and less than or equal to two. With that graph we find the absolute and local maximum and minimum values of the function so well we know that the behavior of the natural logarithms X. Is positive and close to zero. Is that the natural algorithm becomes uh huh very negative. That is a um It takes yeah larger values of magnitude in magnitude and negative and they are negative. So it means that the limit from the right of zero. The functions negative infinity That is we have a zero I'm sorry at one we have natural rhythm of zero and then this function is doing something like this that is is approaching better support coaching the Y axis more and more as X is more and more uh mhm Or is closer and closer to zero. So there is no uh intersection. We know between the line and he grabbed the function that is natural or you're not serious not define but there is always behavior of getting closer to the Y axis, that's for that part. And now we have the natural algorithm is positive when X is heard them one. We have this value of Let's say at two And that value natural Librium of two. It's about C 0.69. That's the value here. And because we have included the altitude in the domain this point here in the graph get to be included and withdraw field or circle. So we have this behavior here and we can see it here will be uh smaller to understand better. Something like this. We had and has been told the vertical institute at um at the y axis that is to graph is a promotion is getting closer and closer to the white adds is from the right never touching it. And that's at one natural logarithms, zero and two. We have a natural rhythm of two which is 0.69. More or less approximation. That's the behaviors a graph that we look here up a little bit bigger. Okay so this is uh behavior and now we can get from this craft that F has on the graph of the function has a hi his point here. So F has an absolute maximum value. That value is natural algorithm Of two which is a presumably equal to 7.69. And that value of course at X equals tomb. That is at the writing point of the interval which is included in this case F has no local maximum because inside the lament there is not a point where which is locally the highest value of the function. So we have this, we have this and we respect to the minimum. There is no local or absolute max minimum. It has no local or absolute minimum. Yeah. And that's because these functions always uh it's always increasing that is mhm. It's always having greater values and is um getting closer to the X axis. So it's never it's never stopping getting closer. It's never stopped getting bigger in magnitude. But negative for that reason there is no small smallest value possible for the function or will always point in the graph. It's always is getting closer to the access and getting the negative infinity. Okay, so these are the properties of dysfunction, natural algorithm of X. For X. Positive and less than or equal to two.

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