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(a) State the First Derivative Test.

(b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?

(a) If $f^{\prime}$ changes from positive to negative at $c,$ then $f$ has a local max at $c$ .

If $f^{\prime}$ changes from negative to positive at $c,$ then $f$ has a local min at $c$ .

If $f^{\prime}$ does not change sign at $c,$ then $f$ has no local max or min at $c$ .

b) We used the concepts of first derivative and second derivative to solve this.

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Missouri State University

University of Michigan - Ann Arbor

Idaho State University

Okay, so we're being has to state the first and second devoted. Um, Okay, So the first derivative test tones is that if a prime is changes from negative, I'm so I would start with parts. Start with positive if prime changes from positive, too negative as my point. See, then it has a local Max has local time out. Call Max at sea. And this makes sense because if you think about it on the graph is increasing. I'm so it's scoring up. A proven there suddenly training to go down. You could see that at this point when it changes, it hasn't had the local Max because it is greater than all the points in the neighborhood. So that's why this is very important. So that's not important to understand visually. So that's what the first point, the second point is that a crime, it's the other way. So it goes from negative two positive on an interval, and it occurs that point C. I didn't have it. Has a local men has a woman? Oh yeah, that's C. And this makes sense because when you go on negative, negative, negative, positive, positive. The local occurs because this is the lowest point in the neighbourhood of this in the neighborhood or surrounding point. And then if ah farm did not change, so there's no change. Then there is no local Max on. Then there is known local maximum. And the reason why this is true, because if you're increasing, you don't know this point is the local Max, because we don't know if it's going up or down, up or down. It's just gonna be keep going up. And it could be a global max. It could be not. Not sure it all depends. So when we know there's no change, we know for a fact that's no local make no local matter, little man. And then for the second derivative test, um, we take the second derivative, which is part of the name. So if trying back is greater than zero on an interval, Hi, so on and terrible I then f fairs. Oh, I'm sorry about that. Just to make that I through. Then on an interval, I then f is Khan cave up so con cave up. So that's what that will be. The shape of the graph and this is the second rate of stances very helpful and helping us understand the shape of the craft. Um And then if it's a lesson zero So the second derivative, his lessons you on the interval I Then it is the opposite Witches con cave down and clanking down Looks like this like an upside down you Oh, con que down. And then the last question last part of the question asked us under what circumstances it inconclusive. And what do you do if it fails? We'Ll It is inconclusive. Whenever pride of X is equal to zero, we don't know if it's in clown tip off the country round. And then so then what we do is we were for to the first drip of cast because the first derivatives tests were always tell us what's happening first derivative. Okay. And if you think about it first, you really don't even need this. I can do everyone test because the first driven and test will tell you that the function is decreasing on descent like just do like that, so decreasing because this is where to decreasing. And this is where decreasing and in if it's increasing. The first report of taps you that this is increasing and this is increasing. So the second day of the test, there's really help the dead help with visualize what's going on and make it a lot easier to draw the picture. But you technically don't even need it. And that's why the first derivative testes so important to you, you really only need the first ever all right.