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Problem 53 Medium Difficulty

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x + \frac{1}{x} $, $ [0.2, 4] $

Answer

Absolute minimum value $2$ which occurs at $x=1$ ;
Absolute maximum value $5.2$ which occurs at $x=0.2$

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Video Transcript

let's find the absolute minimum and maximum values of the function F of X equals X plus one over X. On the closed interval, 0.24. Mhm. First thing we have noticed here is that this function is continuous on the interval 0.24. Because the only value of X for which we will have discontinuity of this function is X equals zero which is not included in this interval. So if it continues the interval is a close interval. So we know F attains its extreme values on that interval. And moreover, we know that those extreme values are attained either at the end points of the interval or at critical numbers of the function. So we got to find the critical numbers of F. And we start by calculating the first derivative of F which is equal to one -1 over x squared Mhm. And we can see that this relative is defined At every point or value x on the close interval 0.24. And for that reason the only critical numbers of these function F are those values of X for which the first serve, A T V is equal to zero. Then we gotta start by solving this equation after relative equals zero. And that's equivalent to the equation one minus one over X squared equals zero. Because the first derivative of F is given by this expression, this equivalent also to one over X squared circle one which also equivalent to X square eagle one. And from this we know that X can be equal 2, 1 or -1. Yeah, but X equal to negative one is not in the close interval from 0.2 24. Then the only critical number of F In the intervals Europe 24 is x equal one. So we got to evaluate the function at this spiritual point x equal one. And at the end points of the interval 0.2 and four among those three values are the extreme values of functions. So Let's start by evaluating f at 0.2. That is 0.2 Plus one over 0.2 which is equal to 0.2 plus Uh 3.2 is to over 10. So this will be 10/2 is the reciprocal of that number. And that's five. So we got 5.2. Now F at four which is c right in point of the interval is equal to four plus 1/4. Just four plus 0.25. And so we get for 0.25 and finally F at the critical number one is one plus 1/1 which is to And so we can see here that the maximum value of the function Over the interval 3.24 will be f at 0.2 which is 5.2. And the minimum value of the function over that interval is two Attain at X Equal one critical point. So we can now write the answer to the problem. Uh huh. The absolute maximum value of F In the interval 0.2 four is 52. And that value of course at the left point of the interval AX equals 0.2. Yeah. And on the other hand we have that the absolute minimum value of f in the interval 0.2 four? Yes two. And that value of course or happens at at the critical number X equal one. So that's the answer to the problem. And then we may make summary. Now we first noticed that this function attained its extreme value over the given interval because the functions continue over the interval and the intervals a close interval. Okay. We know moreover that uh the extreme values there is the absolute minimum and absolute maximum values of the function over the closed interval are occurring either at the end points of the interval or at critical points for critical numbers of the functions. So we got to find the critical numbers of two function for that. We calculate the derivative of F and resolve or solve the of equation F. First derivative of F equals zero. And we got in this case two solutions one a negative one. But we gotta stick with solution that is in the interval 0.24 in this case is X is equal one. And now we have a lot to function at this critical point X equal one. And at the end points of the given interval. And from these three values we found that The maximum, the absolute maximum value is 0.5 attained at The left and point of the interval 0.2. And the minimum absolute minimum value of the function over the integral ship into forest to add or retain it. X Equal one. Which is the only critical number of F in dangerous. And remember we know that this value X equal one is the only critical number of F in the interval. Sure 24 because the other possibility which is negative one, It's not inside or within dangerous Europe 24. So it's the only critical point. A number of F. N is in fact the point or value where the function attained its minimum or absolute minimum over the interval.