A given function has a local maximum at x=-2 \text{ and } x=6 and a local minimum at x=1...

Question:

A given function has a local maximum at {eq}x=-2 \text{ and } x=6 {/eq} and a local minimum at {eq}x=1 {/eq}. Find the concavity of the function at its point of inflection.

Second Derivative Test:

If a function has a local maximum or minimum at a point, we know some information about the second derivative at those points. This is because the second derivative test states that a function is concave down at a local maximum and concave up at a local minimum.

The function has two local maximums and one local minimum. The second derivative test allows us to draw conclusions about the concavity of these three points, even though we don't have any other information about the function.

Since the function has a local maximum at two points, when x is -2 and 6, the function is concave down at those points. Likewise, the function has a local minimum at one point, when x is 1, so it is concave up at that point. If this is a continuous function, the only way that it could switch from concave up to concave down and from concave down to concave up is for the function to have an inflection point in between.

This means that the function must therefore have two inflection points: one between -2 and 1, and one between 1 and 6. The function will begin concave down, then switch at the first inflection point to be concave up, then switch at the second inflection point to be concave down once again.