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Let $ \displaystyle g(x) = \int^x_0 f(t) \, dt $, where $ f $ is the function whose graph is shown.

(a) At what values of $ x $ do the local maximum and minimum values of $ g $ occur?

(b) Where does $ g $ attain its absolute maximum value?

(c) On what intervals is $ g $ concave downward?

(d) Sketch the graph of $ g $.

a.Local minima at $x=4$ and $x=8$

Local maxima at $x=2$ and $x=6$

b.$x \approx 1.9$

c.$(0.8,2.8),(4.8,6.8),(8.8,9.8)$

d.A graph with the characteristics on the left

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Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

they were given the graph of a function F. And were given the function G. Which is the integral from zero to X. Of F. Now in part a were asked what values of X to the local maximum and minimum values of G occur. Yeah. Think about. Well by the fundamental theorem of calculus it follows that the derivative G prime of X is the same as F. Of X. Now notice that at X equals four and eight we have G prime of zero. So G prime of four and eight are both zero. Where you don't you get it from the graph of F. And also notice that just all the slopes are positive. Six. So you my son speakers near X equals four and 8. We have that G double prime of X. Well this is the same as F prime of X, which we see as positive since the slopes are positive and therefore by 2nd derivative test, nine months later that right. Mhm G has local maximum or minima I should say. Mhm. At X equals four and eight. Jesus shit. Likewise, Gucci doctor we have that G prime of two and six while these are zeros of F. And so G prime is zero also Near x equals two in 6. G double prime of X which is the same as F prime of X is negative because the slopes are negative. Mm and therefore by the 2nd derivative test, G has local maxima at the points X equals two and 6. Would you all right now in part B were asked where G attains its absolute maximum value like we said and six all cities. Listen, yeah, we're gonna need you to get a hop on. I just got off. Sure. Really suit well. G is increasing when G prime is greater than zero in decreasing when G prime is less than zero. I scrolled them as more tossed. G increases the most during the first hump which is about full slip. The Interval 019. Yeah, I'm getting slack All the humps after that are alternating down and up and they get smaller in size. So the absolute maximum of the whole interval should be at the end of the first hump. Yeah, so it follows the G has an absolute maximum. It's like more at about X equals 1.9. Yeah, In part C were asked them what intervals the function G is concave downward? Yes. Sign. Various sessions are part it's just part government. Well, the concave down parts of G. These are the intervals where G prime, the derivative of Gs is decreasing. Mhm. We make estimates. It appears that F is decreasing and therefore G is concave downward on the intervals 0.8 About 2.8 but it does look very funny. 4.8 6.8. Sure. No. And 8.8 9.8. Not something you should make. But he also knows twice and never coming in. So basically disrespect. I'm already honestly finally in part d rest to sketch the graph of G. We'll use the previous parts to do this. Yeah. Okay. Second. Already. Just homes with his friends of the jews. I'll notice first of all that G. Of zero is the integral from 0 to 0 of Yeah. A function F. F. T. Which because the limits of the integral of the same is simply zero. And therefore the curve passes through the origin from part A. We know the points of local maximum minima. We also use part C. To draw the con cavity of the graph. And so I'll sketch the graph of G. Here this. Mhm. The internet. Yeah. Mhm. And what they do is that you she's got blood. So we start out at the origin. Yeah. Yeah. And in part C were increasing and we're concave upward until about X equals Uh 1.8. Yeah. Get this sex. We have an abs or a local maximum. Which also happens to be our absolute maximum. Then after that we decrease until we have in Local minimum and about x equals say four. Then we have a local maximum in about X equals six. Which is smaller than the original. The absolute maximum. And we have another local minimum at X equals eight. Which is higher than the other local minimum. And finally we have another local maximum. I guess about X equals 10. Probably kinda like this. She is lower than the other two local maximums. Mm hmm. But So we increased from 0-2. But we're only concave up from zero to about one half. Or maybe not 1/2. But one. I was Yeah. And then we switch, it's concave downward would decrease Until about X equals three. We switched back to concave upward. I'm chuck and I probably should. And then we stay concave upward until about X equals five. Where we switched concave downward again. You just do lots in game. Yes. And then Tell about X Equal seven where we switch back to concave upward. Thanks people like real smith. And then we switched back to concave downward at about X equals nine. She's not sure. And so the graph of G looks something like this. Sure.

Ohio State University