Science:Math Exam Resources/Courses/MATH110/April 2019/Question 06/Solution 1

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:

1. Find the values of x for which g "(x) =0;
3. 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.