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Find the number $ c $ that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at $ (c, f(c)) $. Are the secant line and the tangent line parallel?

$ f(x) = \sqrt{x} $, $ [0, 4] $

\[

f(x)=\sqrt{x},[0,4], \quad f^{\prime}(c)=\frac{f(4)-f(0)}{4-0} \Leftrightarrow \frac{1}{2 \sqrt{c}}=\frac{2-0}{4} \Leftrightarrow

\]

$\frac{1}{2 \sqrt{c}}=\frac{1}{2} \Leftrightarrow \sqrt{c}=1 \Leftrightarrow c=1 .$ The secant line and the tangent line

are parallel.

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Okay. The question is asking us to find the numbers. See, that satisfies the conclusion of the main value through on the givens in Tegel. And then it asked us to graph a function. The secret line through the inn points and a tent line at CFC and is asking us are the Sikh in line and turn in line? Parallel. Okay, So there are two conditions required for the mean very the're, um one is there has to be continuous on the clothes interval from zero four. The function and they function have to be defensible from zero for on the open interval before. So we're called the graph of describe it of X Look, something like this. Um, it looks like a curve. Something like that. And it is defined it zero. So we know it is continuous on, therefore, and you can also take the derivative any point along this confidential. We also know it defensible. And now we're going to be finding the numbers, See that satisfying conclusion. And remember the conclusion of the mean various states that there's some C just at the derivative of the function. The point see into do every of the function. Give just average slope which is or what you can simply find by just doing every four minus zero all over four minus. And now we can solve for this right away. So they recall that the square root of X can be rewritten and X to the one half this final hour. Tink, you're doing much more easily. So the derivative we're gonna be writing in terms of sea which will get us one over to see and that is equal to Now we plug in the values of the script or minus two squared zero all over for you on the square root of four is simply too. And it's great of zero zero. And so it leaves us with two or four, which is the same as one half. And by just looking at this, you can see that a sea of one will satisfy this equation, shall see a tick or two one. And now we're being asked to find the actual equation of the line that is being produced here. So we'LL do that was to the next page. So the way you can do this is you do just to the point slope formula. So in this case, we want to find the Qianjin line at this point. So at FC and then we know the slope, which is one half so Well, put that in and then X minus C because that is the actual explain. So why mine it? And so I called that r C value was one. I got a C equals one. So when you plug in in our f of x equals X and so we can plug it and we can plug in one for this function, So square root of one is just one. So we're gonna play screwed one, a slope we found with one half. And then, since I see is one week, also write X minus one and then we can rearrange this equation on it will give us apply equal one half X, both one. So this is the equation for the tangent line on the graph. Now we're going to find the secret life. So the secret line is also very similar. Way have to do is steam exact things Point slope formula only wine minus after zero. And then we calculate the slope. And the first point is we're just using zero zero as a one point, because in the point, flip formulate could use, um, either point. So, for just simplicity case, we're just going zero zero. Since that is honor Interval. Recall that. And the slope Ashley is calculated by just doing why to minus y one. And since our intervals for Teo I'm zero for We just do. Why? To use it before my geo all over for him. Right? This warrant for minus zero and this is gives us one half that we already knew that. And effort zero squared of zero zero. So this is just this is Theo, and this is also zero. So we could just rewrite this as well. Like corn half X. And if you look at these two ah, formulas, they have the exact same slope except a tangent line. It just shifted by a factor shifted up by one half. So what does this mean on this graph? All it means is that from zero to four, So let's just say this is for this is what it looks like. This is the slope, and then at the value of one was just the ones right here and this equation right here. This line, it's just sdo So this is the white was one half X And then this is why girls wanna have X plus one half and this is the one that's tangent. And so the question here's ours to seek in line and tension lines parallel. And yes, they are because they have again remember the same slope. So I have the same slow, which is amicable one half, but just one of them had just translated up.