Theory and Examples

Sketch the graph of a dif...

Question

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Which of the following graphs has a local maximum at -2.1 parentheses and a local minimum at 5.3 in parentheses? Awesome. So it appears what we're ultimately trying to solve for is looking at our multiple choice answers for A, B, C, and D. We're asked to determine which of these following graphs has a local maximum value at parentheses -2.1 and a local minimum at 5.3 or 3 or 5.3. So these are coordinate points. So, With that said, let's quickly investigate our multiple choice answers to see what our final answer might be, noting that for both, for all of our multiple choice answers for A, B, C, and D, we have our curves represented in blue, and there appears to be several. Peaks and troughs for each of our graphs. And they're all shapes and sizes. So for letter A, the curve resembles almost like a mountain range with a deep dip in the middle, so there's two curves on either side of one big dip, that would be a min value, whereas for our graph for B looks like a mountain range too, but it looks more like waves, because it has two high peaks and two low peaks, whereas our For graph for A has two high peaks in one low peak, whereas for graph C, there's two high peaks in one, and so there's 2 high peaks and 2 low peaks. However, there's a huge gap between the peaks, so where it increases dramatically. And then for D, we have 2. Two troughs, so we have 2 dips and 1 peak. So we have 2 troughs and then 1 peak for our curve for deep. So now we have a visualization of what's happening for this particular problem. Let us recall and note that the local minima and local maxima are the lowest and highest points in the region, respectfully. And if we are given a graph to look at, we need to look for the peaks and valleys. So like I was saying, I said troughs, but you probably heard them as valleys. So peaks are like the top of a mountain and valleys are like the dips, like the troughs. So like the like the inside of a bowl, so like the top of the mountain and then the valleys would be like the inside of the bowl where it dips down. So with that said, We're looking at the peaks and valleys, so the peaks correspond to the local maxima and the valleys correspond to the local minima. So the graph that has local maxima at -2.1 and 5.3 parentheses, and a valley. So once again, so we're trying to find the local maximum, so we're told. Once again, since I said there was a local maxima at 5.3, but it turns out we need to be very specific here. So we're told that the local maxima is at parentheses -2.1, and then we have a local minimum at 5.3 in parentheses. So, once again, we need to note that focusing our attention, so let's look at each graph for multiple choice answers individually to investigate it. So looking at our graph for multiple choice answer A, we could see that this particular graph has a local maxima at -2.1 here and I'll circle it in red. So here's a local max at -2.1, and as we mentioned before, at 5.3. So, Essentially, this is incorrect right off the bat because we're trying to find a graph that has only one local maxima at -2.1, and one local maximum at 5.3. So that will mean that multiple choice answer letter A is incorrect. Looking at multiple choice answer letter B right now. We could see that there's a local maxima at 5.3, near X equals 0, and the valleys, which are the local minima at -2.1 at or nearby, X equals 2, so this means that this is also an incorrect option. So focusing our attention now on multiple choice answer letter C, we can see that it has a local maxima. At -1, sorry, at 2.1, which is good because that means it meets our local maxima requirement. And then now we need to check the local minimum, which it has a local minimum at 5.3. And as you can see, the graph has a peak or a local maximum at -2.1 between X equals 3 and X equals 4. It also has valleys local minima at 5.3 near X equals 1. So this, without a doubt has to be our final correct answer. But just for fun, let us see why multiple choice answer D is incorrect. So multiple choice answer. Letter D, as we can see, the graph has a local maxima, or so let's start with local minimal first. So it has a local minima at -2.1 and at 5.3. And it has a local maxima between, so basically, so the peak, the local maximum that's between. X equals 2 and X equals 3. So as you can see, since these are both minima values, that means that multiple choice answer letter D has to be incorrect. So that means, without a doubt, our final answer has to be multiple choice answer, letter C means it is the only graph out of all of our multiple choice graphs that fits the conditions set to us by the problem itself. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

Key Concepts

Differentiable Function

Local Maximum

Local Minimum

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