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 maxima at (1, 1) and (3, 3)

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says, which of the following graphs have local maxima at the point of minus 3.2 and the point of 2.5. Below the problem, we're given 4 different answer choices with 4 different graphs. So, in order to answer this question, we just need to examine each of the graphs in our answer choice to determine if we have a local maximum at our two given points. So we're going to begin by looking at the graph for choice A. And if we look at the point of minus 3.2, We see that just to the left of this point, that our function is decreasing, and just to the right of this point, our function is increasing. So that means we have a local minimum at the point of minus 3.2. And since we're looking for local maxima, that means that choice A cannot be correct. Now, if we look at choice B, And we look at the point of -3.2. We see that our function is increasing just to the left of that point and it's decreasing just to the right of that point. So that means that we do have a local maximum at that point. And if we look at the point of 2.5, we'll see something similar just to the left of that point, we see our function is increasing, and just to the right, our function is decreasing. So we do have a maximum there as well. So that means that choice B is one of the correct answers. If we look at choice C and look at the point of minus 3.2, We'll see that our function is decreasing just to the left of that point, and increasing just to the right of that point. So again, here we have a local minimum, so that means that choice C is not correct. And if we look at choice D, And we begin with the point of minus 3.2. We see that we do have a local maximum at that point, since our function is increasing just to the left of that point and decreasing just to the right of that point. But if we look at the point of 2.5, we see that this point does not fall on our curve and therefore it can't be a local maximum. So that means that choice D is not correct. So the only correct answer choice for this problem was answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Theory and Examples

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

Key Concepts

Differentiable Function

Local Maxima

Graph Sketching

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