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Point) Let f () IO) . List the critical points in order from smallest to kurgest:Wnte INC or DEC t0 describe if the function is increasing or decreasing on each int...

Question

Point) Let f () IO) . List the critical points in order from smallest to kurgest:Wnte INC or DEC t0 describe if the function is increasing or decreasing on each interval: 6,4)(A,B) (B,o)Determine if the function has & local maximum (MAX) or minimum (MIN) or neither (NEITHER) at each aiical ptit

point) Let f () IO) . List the critical points in order from smallest to kurgest: Wnte INC or DEC t0 describe if the function is increasing or decreasing on each interval: 6,4) (A,B) (B,o) Determine if the function has & local maximum (MAX) or minimum (MIN) or neither (NEITHER) at each aiical ptit



Answers

Use the graph of $f^{\prime}$ to (a) identify the critical numbers of $f$ , (b) identify the open intervals on which $f$ is increasing or decreasing, and (c) determine whether $f$ has a relative maximum, a relative minimum, or neither at each critical number.

For this problem, we have been given the graph of a function. We have been asked to find the critical numbers Associated with this graph and determine if there are relative or absolute minimums or maximums at these points. Now, let's review what a critical number is. But critical numbers happen where the derivative of my function is zero or it's undefined. X. Think about what those mean. Undefined often means that we have some kind of point or cusp or it comes up to a point or down to a point. We have that sharp curve or a spot where maybe there's a disconnect, there would be undefined at that point as well if my derivative is zero, that's often a minimum or a maximum point, because that means my slope is zero. So we're looking for places where the slope is zero where it's undefined. So let's take a look at this problem here. This looks like a parabola. And if I was going to approximate the point on the problem, I would say that's X equals two and y equals four. So that would be that point where the slope levels off. I have a slope of zero. That's that first case on the function. So what kind of the point is this? Well, that is the highest value of the function on my graph here. So this is an absolute maximum. Now this graph does not have any absolute minimums because it's an open graph. If you look at as this graph comes down on either side, you can see those open end points at the bottom right where it hits the X axis. So the value of my function is going to get closer and closer and closer two, the X axis, but it's never going to hit there. So we this would not have a minimum value, It just has the absolute maximum.

For this problem, we have been given the graph of a function and we've been asked to approximate the critical numbers of this graph. And then we're going to identify are they absolute or relative minimums and maximums? Or none of those. So let's take a moment review what a critical number is. If I have a function F of X, Then the critical numbers are going to happen when either the derivative of my function equal zero or the derivative is undefined. So what does that look like if I'm looking at a graph? Well, if the derivative is zero, that means it has a slope of zero. So if I imagine drawing a tangent line to the graph, that's going to be where that tangent line, is horizontal slope equaling zero undefined means I don't have um I don't have a slope. So possibly it's a cusp point. Um You know, maybe it comes to a sharp point like that, something like that. Perhaps, maybe there's a point where there's a disconnect on my graph and I don't have a slope at a given point that could also be a critical number that we would need to look at. So for this particular graph there are no cusp. So I probably don't have any cases where my functions under the derivatives undefined, but I do have a place where the slope is zero and that happens at the point X equals zero, Y equals zero right there at the origin. Now, is that origin a minimum or a maximum? Actually it's not because the function continues to increase from the entire range from executing negative one, positive one. So this is not a minimum or a maximum, it is none of the above.

For this problem, we have been given the graph of a function. We've been asked to approximate the critical numbers of this function and they were going to classify them minimums, maximums or none. So, first, let's remember what a critical number is. Critical numbers. If I have a function f of X, critical numbers are going to happen in one of two places either where the derivative of the function is zero or the derivative is undefined. So what does that look like when we're looking at a graph and not necessarily the equation of the function? Well, if the derivative is zero, that means I have a slope of zero. Or if you can imagine a tangent line means a tangent line is going to be horizontal. The other option is the derivative is undefined. That often happens where you have a cusp. Maybe it's going down like this, up to a peak. There's something going on where I don't have a defined slope at that point. So let's take a look at the function we have here, We do have a cusp. We do have a case where we have, the derivative is undefined and that happens at the .20. At that point, I do have a cusp. So that's going to give me an undefined derivative. I also have to places where I'm going to have, if I imagine a tangent line, it would be horizontal and that's at the top of those two peaks. One happens when X equals equals one, so that's the 10.13 And then again when X is three, so that's the 30.33 So three points. Now let's classify them first. Let's start with um The Cusp that undefined point at 20. That is the lowest point on our graph. Okay. Nothing is right there on the X axis, there's nothing below that. So this is a minimum point. It is definitely a relative minimum because everything around it is bigger. This is the lowest point locally, but it's also the absolute minimum because nowhere else on the graph, is there any point lower than this one? Okay. How about the blue points? The tops of those peaks? Well, those are maximums and they look like they have. If I'm eyeballing this, they look like they have the same value. So I would say these are both maximums. They are relative maximum, certainly because everything around them, the value, the function is less, but nothing anywhere on this interval has a higher value. So they are also absolute maximums. So three critical points to maximums and one minimum.

For this problem, we've been given the graph of a function and we've been asked to approximate the critical numbers for that function. So let's take a moment and review what a critical number is. Critical numbers happen if I have a function F of X. Critical numbers occur when either do the derivative of the function equals zero or it's undefined. So when we're looking at a picture of a graph of a function, what do these correlate to? Well, an undefined derivative, It's going to happen often at a cusp when it comes to a point, either up or down. Maybe there's a disconnect somewhere. There is some place where I do not have a derivative. So those would be undefined for the derivative equal zero. That means that the slope is zero, which means that I have a horizontal tangent line. So those are the things we're going to look at for this problem. So if we look here, there are no sharp points or disconnect or anything like that. So we do not have a spot where are derivative is undefined. But there are some spots where if I imagine a tangent line moving along my function, there are actually two places where that tangent line would be horizontal. The first one is at approximately X equals two. Okay, so that's where the function levels off. Now at this point that is not a minimum or a maximum because the function is increasing from the beginning. All the way up to the peak at about X equals five. So this is it's a nun. It is not a minimum. It is not a maximum. We do have a maximum though at X equals five. That's our other critical point. At the very peak of that top, That's where the horse. The tangent line to be horizontal has a slope of zero. Which means that's a critical number and that is an absolute maximum because it is the highest point of our function.


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