Science:Math Exam Resources/Courses/MATH110/April 2019/Question 06

MATH110 April 2019

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Question 06
Consider a function $g(x)$ that is continuous and differentiable everywhere. Assume that $g(x)$ has ALL of the following properties: $\lim _{x\to \infty }g(x)=-\infty$ and $\lim _{x\to -\infty }g(x)=0$; $g'(x)>0$ for $x<0$ and $g'(x)<0$ for $x>0$; $g''(x)>0$ for $x<-1$ and $g''(x)<0$ for $x>-1$. Sketch the graph of $y=g(x)$ and indicate where the inflection point(s) occur.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

The first bullet point tells us what the function looks like for x small enough or large enough: g(x) has a horizontal asymptote as $x\rightarrow -\infty$ and it should look go down to $-\infty$ as x increases. From the second bullet point, we can infer that g has a maximum at x=0. The graph of g away from x = -1 looks something like the graph of $e^{x}$ for $x<-1$ and the graph of $-x^{2}+1$ for $x>-1$. A partial sketch is presented in figure 1. Note that there are many possible different sketches. Figure 1: Partial sketch of a function g(x) satisfying the properties mentioned in question 6.

Hint 2
A point x is an inflection point of g(x) if $g''(x)=0$ AND the concavity of g changes across x. A function is concave up at a point x if $g''(x)>0$ and similarly concave down if $g''(x)<0$. This should also help with completing the sketch of g.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Figure 2: Sketch of a function g satisfying the conditions of question 06. Recall that x is an inflection point of g(x) if g "(x)=0 AND the concavity of g changes accross x. The problem is split into these two parts: Find the values of x for which g "(x) =0; See the concavity above and below of x are different. From the third bullet point in the statement of the problem the only possible value where g "(x) =0 can happen is x= -1. Assuming that g "(x) is a continuous function this is indeed the case by the IVT. Moreover, g "(x)>0 if x< -1, which means the function is convex up in this interval, and similarly g "(x)<0 if x> -1, which means the function is convex down in this interval. Therefore, the function changes concavity across x=-1 and this is an inflection point. A final sketch of the graph is shown in figure 2.