3. Recall that the Extreme Value Theorem guarantees that continuous functions have global maxima and global minima over every closed, bounded interval:Consider the ...

3. Recall that the Extreme Value Theorem guarantees that continuous functions have global maxima and global minima over every closed, bounded interval: Consider the following mathematical statements. Fill in the blank with "all" "no" or "some" to make the following statements true Note that "some" means one Or more instances, but not all If your answer is "all" then give brief explanation as to why: If your answer is "no" then give an example and a brief explanation as to why: If your answer is some" then give two specific examples that illustrate why your answer it not "all" or "no" Be sure to explain your two examples_ For real numbers b, if f(c) = 22, then f has a global maximum on the interval (0,6)- For functions f , if f is differentiable and has global maximum on the interval 0 < € < 10, then f' (#) = 0 for some € in the interval (0,10). For functions f, if f is continuous and differentiable on 0 < € < 5 and f has exactly one critical point at € = 3, then f has either a global maximum Or minimum at T = 3_

So by definition of function is continuous has to limit from the left ankles. The limit from the right exists and therefore it also equals half of half of a. So they from the left from the right equals f of a right. So if you take a look here, you could see that half of zero is equal to zero. But the limit as X approaches zero from the left is going to be negative infinity and serve from the right is equal to infinity. So these are not equal and it doesn't equal zero. So therefore ffx is not continuous At X equals zero. Okay, take a look at the next part here. So for part B it's asking so show that it doesn't retain naps and max are meant well. So we know that uh half of thank you to everyone is equal to -1, half of one is equal to one. And then it goes from negative one all the way to infinity as it gets closer to zero and it goes from one to infinity, right? So because it goes from negative infinity infinity and infinity and everything basically here we've got that the range of values goes from negative infinity to negative one. Union zero. The union want to infinity. Okay. And so this basically means that there is no absolute max from it. Okay, So for port, see asking for a sketch. So that would be this Got a point at 00. It's gonna make that or try to read here. I look better. And then we've got point and negative one goes down really there 0.1 and separate like this, this one say that one that's the sketch.

In this moment. For differential ability right X equals to zero. You must have the left hand side of the limit, which is LTD let's dance studio. Therefore zero. My message Minour have zero by my message. This is the thing about the limerick that's tends to zero. This is my research. Keep on -1 by Edge minus epa one birds. You got it. Bye minus age. You found -1 by age. Let's keep our one by which this becomes limit acceptance to zero. You pound -2 batch -1 on the par minus too much. That's one this is minus in life plus one inch chest minus one. Similarly the right inside of the derivative limit is going to ask him it. That stands to zero airport zero plus set -F0 by yeah putting the values similarly we get this is one -U. part -2 birds upon one plus Yvonne -2 virus, which is one x 1 which is one. You see that two limits are not equal. It means that affects is not different people that X equals to zero. But since the limits are finite, therefore effects is continuous at X equals to zero. So I fix this continues everywhere. But not the friendship genetic sequence. Yeah.

Continuous function is equal to X. Taking the first partial derivative yes, absolutely. X is equal acts and f sub y z zero using the same partial derivatives. You know that upset X X is equal to one. I've said Y Y is equal to zero f sub x Y z zero and therefore the discriminate Izzy zero. So they're what the second derivative test is going to be inconclusive. In fact, there are no critical points considering this function over its domain. We're going to see that F is in the strip form between the two lines of y is equal to negative X and lies ableto one minus X. Therefore, we can make the function arbitrarily large within this region. In fact, we can see that the limits as X approaches negative Infinity of f is arbitrarily large. This does not contradict the serum in the tax because the Domaine de is about a domain on in that for any interview, and we can see that the open interval off negative and comma and plus 0.5 is contained within this region.

In this problem we have ethics equals two submission And equals to 0 to infinity. Expo in my in fact Odeon. Love of a power mm This can be written down some nation And equals to 0 to infinity. Ex lover following and by and factory which is nothing but E power X log of air. It's just you followed off. Yes. And at this point left and certain limit for the detonator. This is limit That stands to zero. I felt 0- at minus because you don't learn minus a the scrums us limit. That stands to zero. A pawn minus. It's by my minus one by my message which is log into the base. E similarly that are concerned with the limit is Limit extends to zero assholes. Zero plus it's minus. I have zero words. This is the number Which tends to zero. Your brother -1 by that's organise it's busy and we see that you are equal. It means that if X is differentiable attacks equals to zero Since every differentiable functions continents FX is also continues our tax equals to zero correct options are that's all