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Problem 26 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) = e^x $


No extrema.


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Video Transcript

we will sketch the graph of the function exponential backs and use that graph to find the absolute local maximum and minimum values of the function. So we know that this exporting cell function is equal to one at zero. It is increasing always increasing. It's always positive. It increases quickly. And when X is large but negative, that is that is if we moved on the graph to the left to admit negative infinity. The values that function are closer and closer to zero but always positive. It means that the graph of this function more or less like this. We have an a synthetic behavior to the left and then we have an increasing rapidly increasing behavior to the right. Here we have the value one at 0. So this is more or less the graph the exponential function And we see clearly here that limit. When x goes to plus infinity of the function is plus infinity. It means the function grows without any band when X is large and positive. It means that the craft goes up. When we moved to the right of the graph, the graph goes up and that without any bomb. And the other property is that the limit when x goes to minus infinity If zero. And that means that the function is getting closer and closer to zero. When we move to the left of the graph. That is when we make eggs large but negative, mm hmm. Mhm. And here we are. We are considering the function defined on the whole real numbers because there's no other specification in the statement of the problem. So, we are going to be considering the whole real numbers as the domain is function and we have them these two properties. Mhm. Then with that we can say that there is no hi his point in the graph because the graph is always getting up to the right and because it has an a synthetic behavior near zero, That is approaching the value zero when X is a large negative, we can say that when we move to the left of the graph, the function is always approaching zero. That is, it does not have as as slow as possible value for that reason, for those reasons we can say that F has no absolute extreme. That is no work to the maximum, no absolute minimum at all. And for a similar reason that is there is no point where that can be considered the highest value near or locally and the same thing for the minimum. So the F has no local extreme A at all. So we can say the two things together that this function has no drama at all. And the reason, the main reason for that is his behavior here of always getting closer and closer to zero when we moved to the left and growing up without anyone without anyone with when we move to the right and the function is always increasing positive increasing function. And so the consequence of society that that the function has no stream a at all. and that's the exponential function.

Universidad Central de Venezuela
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