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(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.

(b) Use calculus to find the exact maximum and minimum values.

$ f(x) = e^x + e^{-2x} $, $ 0 \leqslant x \leqslant 1 $

(a) SEE GRAPH

The absolute minimum value of $f(x)$ is $\approx 1.89$.

The absolute maximum value of $f(x)$ is $\approx 2.85$ ;

(b) Absolute minimum value is $2^{1 / 3}+2^{-2 / 3}\approx 1.8898815748 $ which occurs at $x=\frac{1}{3} \ln 2$.

Absolute maximum value is $e+e^{-2}\approx 2.8536171117$ which occurs at $x=1$.

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Campbell University

Harvey Mudd College

Baylor University

Idaho State University

in part A We want to estimate the absolute maximum and minimum values of the function exponential of X plus exponential of negative two, eggs on the interval zero. Want closed interval using the graph of the function and the party we calculate the exact maximum and minimum values of F. Over or on the from the interval 01. So this is a graph of the function. We can see that we have an absolute minimum around this point and the absolute maximum is around this point. So the extra maximum is attained at X Equal one. As we can see here. And there is some point around here where the function attains its absolute minimum. So if you make some zoom e of this graph at those regions there. So we found for the absolute minimum, let's say uh draw vertical and horizontal dot line To show the absolute minimum point. And we can see that the value of that minimum is around 1.89. Okay. And that minimum decor about this value here which is eggs approximately equal 0.23. That's for the absolute minimum value of the function. And for the absolute maximum value, we have this some here around the Mhm. And 0.1. And we can see that the absolute maximum value of the functions about. We can see here It's about 2.85 At Mexico one. In this case we are sure that the absolute maximum value is obtained at Mexico won this case. We know that the absolute minimum value is obtained uh next approximately equal to 0.23. So these are approximations to the absolute minimum value and the X. Where it of course. But now we want to calculate that precisely in part B. And for that we use calculus. So the first thing we got to say is that if function is continuous on the closed interval, it attains its extreme value. That is there are points in that interval where or such that the images of those feelings are equal to the maximum and minimum values of the function over that interval. That's the first thing we know the function attain its extreme values over a close interval, continuous function like this. And the other thing is that we know that those extreme values are attain at either the end points of the interval or at critical numbers of the function in the interval. So we've got to find the critical numbers of dysfunction two give away the function there. And at the end points. And from those images we choose the largest and the smallest. So we're going to find the critical numbers numbers of F. In sierra one For that we find the derivative of f. And that is the same exponential function minus to eat to the -2. X. And that derivative exists for every value X. In the interval 01. And I say that because it's this is true, it implies that the only critical numbers of the function are those values of X. For which the derivative is syrup. So the only critical numbers of F. Our does values of X. Voyage Derivative is zero. So we gotta solve this equation, derivative of F equals era. So that very difficult. zero is the same or is equivalent to E. To the X minus two. E. To the negative two eggs equals zero. And that's equivalent to E. To the X equal to E. To the negative two X. Which is equivalent to eat reacts equal to to over eat two eggs. We passed this denominator here to the left and that's equivalent to E. to the three eggs equal to in applying natural algorithm. And taking into account the natural algorithm is an objective function. We have again uh equivalence with these equation. Natural rhythm of E. To the three eggs Equal to the natural rhythm of two. And he is equivalent to uh three eggs. Because the natural algorithm and the exponential functions are inverse to each other. Equal natural algorithm of two. And this is the same as saying that eggs Is the natural rhythm of 2/3. Yeah. And that's the exact value word, the derivative zero. But now we can use a calculator to find that this number is about 0.23 104 90602. So we have this solution. And as we can see this value is inside the interval 0 1. So it's the only yeah critical number yeah of F. In Tijuana. And in fact in the whole real numbers threat In 01. That's what is important in this case. So we have found all the critical Numbers. That is the only one that exists for the function in the Industry one. So we got to evaluate the function now at These critical point. This critical number and the end points of the interval that is the number zero and 1 Then F0 is exponential of zero. Last exponential of -2 times zero. There is too. Or if you want to be clearer is either zero plus peter zero. That is F. at one is E. To power one plus eat the negative too. Or E plus E. To the negative two. And it's in a calculator. This is about mhm two point 85 361 seven 1117. And F. At the critical number. Natural rhythm of 2/3 is equal to E. To the natural logarithms to over three Plus E. to the negative to natural rhythm of 2/3. And we can write this as E. To the natural logarithms of two to the one third plus E. To the natural liberalism of two. To the negative two thirds. And we know this is equal to two, the one third plus To the native 2/3. And that's approximately equal. It's in a calculator to 1.889 88 one 5 7 for eight. So we have these three images here. And among those images are the extreme violence as a function. Because we know that the continuous function attains its extreme values over close interval. And those points or numbers in the close interval where the function obtains those extreme values must be either critical numbers or the end points of the internet. This case we have only one critical number inside the interval 01. And so we have only three options. The images of the end points of the of the interval and the image of the only critical point of F. And we can see that this is Than the maximum values 2.8536171117. And the minimum value is 1.88988157 for eight. So we can say that F has an absolute minimum value. Um 2 to the 1 3rd Plus 2 to the Native 2/3. This is exactly the exact minimum value of the function over the interval one as your one. And we can say soo wan we put it in here and we know that this value is aboriginal. The equal As we saw above to 1.889881 5748 owns here one. And If that is the absolute minimum value of the function over the interval 01 which is this value here. And this is an approximation to that value is obtained at uh the only critical point of the function F. So I would say that year. Mhm And eat. Of course. Yeah. Yeah of course add de critical number Eggs Equal Natural Liberty the most to over three. Then we have that F. Has an absolute mean maximum. Yeah. Value That that's the um that's the maximum is 2.85 which is E plus E to the negative too. So it's the exact value E plus E. To native to. And it's around one for 2.85 3617 1117 announcer one. I need a course add the the writing point at one that is the writing point except for one. So we have this uh Results and we can see that we have we are in accordance to the estimation we didn't parade because the minimum is around 1.8 nine. Is we around this number 22 decimals. And of course that natural rhythm. Just two of the three which is about 0.23 that we saw here after the minimum critical number Where the relative ease zero And the value of the absolute minimum is around 1.89. And of course at about 2.23. That is correct. And for the absolute maximum it of course at the writing .1 and it's about 2.85 as we saw here. So it's a in accordance to what we did in for a but here we have the exact values that is the absolute minimum and the absolute maximum in a close mathematical form, two to the one therapist to to the negative two thirds for the absolute minimum value, and E plus E to the negative two for the absolute maximum value. And the points or numbers, were those extreme occur are also uh exact because we have for the absolute minimum value X. Is X, where courses Natural Liberty um of two of the three. And for the absolute maximum value, of course at X Equal one.

Universidad Central de Venezuela