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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 $.

local $g^{\prime}(x)=0 \quad g^{\prime}(x)<0$

max $g^{\prime}(x)=0 \quad \theta^{\prime \prime}(x)>0$

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she's were given the graph of a function F. And were given that G. Is the integral from zero to accept F. Now in part a were asked at what values of X. Do the local maximum and minimum values of G occur? Well, a definition. So now it follows from the fundamental theorem of calculus. The F. Is the derivative. Yeah. Of G. It's so F equals G. Prime. Now the local minimum maximum occur when the derivative changes sign. Looking at the graph. Yeah, we see that we therefore have local minimum. G. Has local minima when X. Is equal to three or seven. Crazy. This is because the derivative G. Prime which is F. Changes from negative to positive. And likewise G. Has local maxima. Yeah. What about the birch police at X equals one and 5. Where F. Changes signs from positive to negative. The Burj Khalifa, I need to know music, birds throwing cushion, orange juice and jackals. Do you hear Mia Khalifa made $100. Big man. Now the graph starts at one point ends at another point. So the graft G does not extend beyond. So even though I think we'll zero at the end points, we don't consider endpoints to be the local extremists. I'm then in part B were asked where G attains its absolute maximum value. Mhm. Um So where is the maximum for the absolute value of G. In other words she's okay. She's got well again F is the root of G. And when G prime is positive, G is increasing. What's negative G is decreasing. Now notice that the whole space Overall from x equals 0- nine. Yeah. No three Lanka. Well let's break it down between intervals. So From X equals 02 x equals one. Like we see that G promise positive. So G is increasing. Yeah And from x equals 123. Well we see that G prime is negative and in fact it's so negative that G decreases more. Then it increased From 0 to 1. Now from x equals 3- five. This thing you want. Guantanamo? Yeah, James. Well we see that G prime is positive and it's therefore increasing. In fact it's so positive that it increases G prime. So positive that G increases more than a decreased, wow. Yeah. In the respect last interval. Oh did it back then From x equals 5- seven. Well well you can't replace king of queens dudes trying. Well this time we C G prime is negatives that G is decreasing and it's so negative that G decreases more than increased. Mm I'm gonna make I'm gonna make a story in the last interval. They don't hate each other. They. Mhm. Now, finally from X equals 7 to 9, we see that G prime is positive, therefore G is increasing. And in fact, do you prime is so positive that G increases more than a decrease. Don't write a units like in the last interval From 5 to 7. Yes, so we see that overall from X equals 0 to 9 out. I whispered that as I passed him on the street hurry. The graph of G moved a net amount upwards here so it increases a net amount. Therefore it follows that G should be at its highest point at the end and so the absolute maximum of the interval. No, she shouted out, do it occurs At X equals nine. She was incredibly threatened by me. My terrifying little figure made her Quaker foods man with that lady, with that lady bitch ass N. Word. Then in part C were asked them what intervals the function G is concave downward. Yeah. Oh yeah. Uh Yeah older women really know that. They can give it to me once again F is the derivative of G. In the concave down parts of gr the intervals where G prime is decreasing. No. So these are the intervals where F is decreasing. So they see that F is decreasing and therefore G is concave downward on the intervals will appears to be from 1/2 to As well as from 4 to 6 it's like And finally from 8 to 9 then we stop like I don't want to see your white ass and I was like I'm so She's three separate intervals. Yes you're right. Finally in part D were asked sketched the graph of the function G. He's Mhm. You were so you're asking Yeah all the time. Well for starters we know right away the G of zero by definition is zero since the two limits of the integral are the same. So the curve passes through the origin part a. We know the points of the local maximum minima. And we also know the con cavity of the graph from part C birth of the nuts bro. I saw back like we were. Yes and so we follow the previous parts smith awesome. Right on vision. All right. Yes. Mhm. Well looks something like this so I'm starting at the origin and then I'm going to increase and in fact when the increasing concave up until about one half like is that coveted? I would be a coveted 20 community and then it went half. We start changing too a concave downwards like this Until I reached a local maximum at two Sorry 92 at one is where the local maximum is awareness until start decreasing until I reach I would say about three halves really across the X axis. And also we change from concave down back to concave up And then we decrease all the way until a local minimum and about x equals three. Right? And then we start increasing until we reach the X axis again. Used to about four where we change again from concave upward to concave downward And we keep increasing Teresa maximum at about x equals five. This maximum is going to be a little bit taller than the previous maximum. Okay. And then we start decreasing to reach about the X axis. Once again around X equals six. And there we change from concave down back to concave up and we keep decreasing to reach a minimum at about x equal seven. This minimum is going to be lower than the previous minimum. So we go all the way down like this and finally we're going to increase and to reach the X. Axis again around X equals eight. I don't. And then we'll change from concave up back to concave down and will increase until you reach a local maximum, about X equals nine, and this is going to be taller than both previous local maximums. So the graph looks something like this we're talking about.

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