Problem 8 Medium Difficulty

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = e^x - 0.186x^4 $

Answer

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Video Transcript

we want to produce graphs of F of X is a brute heated X minus 0.186 times X before ever bill all important aspects of the curve and in particular, we want to use the first and second derivatives to estimate the intervals for the functions increasing, decreasing on any extreme values, intervals of con cavity and possible points of inflection. So I went ahead and already plotted a, uh these three graphs. So the first draft for FX is just what I got when I plugged it into my calculations. So I mean, this interval looks pretty good, and we kind of notice that it kind of dips up the dips back down. So let's figure out what window we want to include to get that next part in there for our Y axis. And maybe we need to still go further out left and right as well. So I went ahead and on the first and second derivatives already just for the sake of time. And what I wanted to do was to look for all the X axis. Uh, I mean all the ex intercepts for the first and second derivatives, so I can find where these functions are greater than zero, and less than so for the first derivative. Remember, when this is strictly larger than zero, this tells me where the function is increasing. So let's go ahead and figure out what these intervals are. So to the left of 2.474 it looks like F prime is going to be positive. So that's gonna be negative. Infinity to 2.474 union with and then it looks positive. Begin after 3.597 That goes to now. We know if Prime of X is strictly less than zero, this will be from the function is decreasing, and there's only one in trouble. For that, it's gonna be 2.474 four a little bit better. So 2.474 to 3.597 Now we can go ahead and say whether these Air Max is or men's so into our first X intercept, the functions positive, so it should be increasing into the point, and then it's decreasing afterward. So that tells us this will be a local Max. And for our other Exeter's well, it's negative so the function would be decreasing into the point and then increasing after. So this year should be at local minimum for the next one. If we want to find where the functions conchita, well, we need to find where Double prime of X is strictly a large men's room. And so this tells us Khan caged up so that looks like it's going to be from negatives are 0.562 1.171 and then after 3.15 now for the function being conk it down, it will be where f double prime is strictly less than zero, is it Not a little. And so this is going to be to the left of our first X intercept. So negative infinity to negative 0.56 union with and then from 1.17123 point 15 And now we can come over here and see that we are going from negative deposit. So we are changing from cavity for our 1st 1 So that will be appointed infliction for the next woman going from positive negative. So that is also a inflection point. And then for last we're going from negative to positive. So it would be a point of inflection. So now that we have these five values of interest, I went ahead and planted them so I could figure out the, um why values further. And so these here, remember, are going to be our Max Men's or local necks is immense, and these next two are points of inflection. So I decided on my ex viewing window because I came over here and I saw my largest value for X was about 3.6. So I knew I wanted to be slightly to the right of that. So I just went ahead and chose for and then my smallest value was about negative half. But I went a little bit further to live because I also knew from at first graph that this here should have another X intercept. So I just went ahead and went a little bit further to the left of my ex intercept two. And sure, I get that in there as well. And then for the y values. I did something similar. I looked for my small son, my largest values, so my largest value was about 4.5. So I just went slightly above that to five. And then my smallest value was about 1/2. But I wanted to also include that. Why intercept over there? So I just went ahead and went slightly before, below the ex access to negative. So that was my rationale behind my viewing window and these five points along the graft or the points that graph our maximums and points and infliction. So if you were to just look at it, you might think the only thing of real interest without these points was would be in this upper lodger right here. And you may have overlooked that these two points are actually inflection points, especially from the first crack, because it kind of looks like it just is a straight line from about negative 1 to 2 minutes raises up and then falls back down. So if you were to just look at the graph without figuring out where the points of inflection are estimating that you may have overlooked those two points