Let f be a function that is even and continuous on the closed interval [−5,5] .

Question

The figure above shows the graph of  , the derivative of a twice-differentiable function  , on the interval [–3, 4]. The graph of  has horizontal tangents at x = –1, x = 1, and x = 3. The areas of the regions bounded by the x-axis and the graph of on the intervals [–2, 1] and [1, 4] are 9 and 12, respectively.

Find the x-coordinates of all points of inflection for the graph of  . Give a reason for your answer.

Answer 1

From your description, I'm going to guess that f '(x) looks like this:

Possible POI occur when f ''(x) = 0 which occur at x = -1, x = 1, and x = 3 (occurs when the slope of the f '(x) graph is zero or at local extrema).

To determine if any of these points are actually POI, we need to determine if f ''(x) changes sign (goes from positive to negative or from negative to positive). We can determine the sign of f ''(x) by looking at the slope of f '(x).

For x = -1, the slope to the left is negative and the slope to the right of it is positive therefore x = -1 is a POI

For x = 1, the slope to the left is positive and the slope to the right of it is negative therefore x = 1 is a POI

For x = 3, the slope to the left is negative and the slope to the right of it is positive therefore x = 3 is a POI