Sketch the graph of a twice-differentiable function y=f(x) tha...

Question

Sketch the graph of a twice-differentiable function y=f(x) that passes through the points (-2,2), (-1,1), (0,0),(1,1), and (2,2) and whose first two derivatives have the following sign patterns.

patt

Accepted Answer

Welcome back, everyone. In this problem, which of the following graphs have the first derivative of FX greater than 0 from the intervals -3 to 0 and 3 to infinity, and the first derivative of FX to be less than 0, from negative infinity to -3 and from 0 to 3. Now what are we trying to figure out here? Well, we want to basically figure out where our first derivative is increasing. That is where it is going to be positive, and where it is decreasing, that is where it is going to be negative on our graph. And we can tell that by looking at the slope of our curve, OK? Now if we think about it here, essentially what we're saying if we were to do a quick sketch. And if we're at the hour intervals of negative infinity, -3, 03, and then to infinity. Then basically what we're saying is that we want to find a graph where it is increasing from -3 to 0. OK, let me just put that here from -3 to 0 and from 3 to infinity, but decreasing from negative infinity to -3 and from 0 to 3. So if we look at our choices here, which one of these graphs will fit the criteria? Well, let's start with option A. What do you notice? Well, if we take our intervals here, so we're going from -303, and then our infinities, our infinity values on both sides, OK. Notice that the graph for option A is increasing from negative infinity to -3 and from 3 to infinity. On the other hand, it is decreasing from -3 going to 3. So here we can tell that answer choice A is not the correct option. Let's go to answer choice B. What do you notice? Well, no, notice that for option B, our graph is increasing from -3 to 3, OK? And then it is decreasing from negative infinity to -3 and from 3 to infinity. So in this case, we also know it won't fit the criteria. Now if we go to answer try C. OK. Then what do you notice? Well, no one notice that the graph for option C is increasing from negative infinity to -3 and from 0 to 3, while it is decreasing from -3 to 0 and from 3 to infinity. So neither does C fit our criteria. The horrible answer tries deep. Well, no, for graph D, notice that our function is increasing from -3 to 0 and from 3 to infinity, OK? While it is decreasing from negative infinity to -3 and from 0 to 3. Thus, if we compare to what we're looking for, we can tell that answer choice D is the correct option. Thanks a lot for watching everyone. I hope this video helped.

Sketch the graph of a twice-differentiable function y=f(x) that passes through the points (-2,2), (-1,1), (0,0),(1,1), and (2,2) and whose first two derivatives have the following sign patterns.

Key Concepts

First Derivative Test

Second Derivative Test

Critical Points and Inflection Points

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