FORM GR1268 Question 25

Question:

The graph of the derivative \(h'\) is shown above, where \(h\) is a real-valued function. Which of the following open intervals contains a value \(c\) for which the point \((c,h(c))\) is an inflection point of \(h\)?

(A) \((-2,-1)\)     (B) \((-1,0)\)     (C) \((0,1)\)     (D) \((1,2)\)     (E) \((2,3)\)

Answer:
(A)

Answer Key:
An inflection point is where a curve switches from concave to convex, or vice versa.
A curve is concave if \(h^{\prime\prime}\lt0\) and convex if \(h^{\prime\prime}\gt0\).
At an inflection point, \(h^{\prime\prime}=0\).
Thus, we are looking for a point where \(h'\) is flat (the slope is zero).

There is such a point in the interval \((-2,-1)\).
Since \(h^{\prime\prime}\) changes from \(\lt0\) to \(\gt0\), \(h\) changes from being concave to being convex.

Alternatively, you can use integration.
The area between \(h'\) and \(x\)-axis is initially decreasing.
From \(x=-2\) to around \(x=-1.5\), the area decreases at an increasing speed.
From around \(x=-1.5\) to around \(x=-0.5\), the area decreases at a decreasing speed.
From around \(x=-0.5\), the area increases at an increasing speed.