4. Concavity and the 2 nd Derivative Test

Concavity If we know that a function has a positive derivative over an interval, we know the graph is increasing, but HOW is it increasing? At a constant rate? At a decreasing rate? At an increasing rate? The 2 functions below both increase, but they bend differently. The graph on the left increases at an increasing rate and the second increases at a decreasing rate. (functions that increase at a constant rate, are linear, boring, and don’t require calculus).

Concavity We can analyze the tangent lines in each of these cases. In the graph on the left, the tangent lines are below the curve and are increasing from left to right. In this case, we say the graph is concave up (like a smile) In the graph on the right, the tangent lines are above the curve and are decreasing from left to right. In this case, we say the graph is concave down (like a frown).

Concavity

Example 1 List the open intervals on which the graph is concave up (CU) and concave down (CD)

Concavity test Anytime we talk about something changing, we’re talking about the derivative When we talk about the slope of the tangent lines changing, we’re talking about how the derivative function is changing – this means we’re talking about the second derivative! Concave up where y’ is increasing (y” > 0) Concave down where y’ is decreasing (y” < 0)

Example 2 Sketch a possible graph of a function with the following conditions

Point of Inflection (POI) A point (c, f(c)) is called an inflection point if the graph of f changes from concave up to concave down OR from concave down to concave up. This means that f’’ will change from positive to negative or negative to positive. Concavity can also change at a discontinuity, but it won’t be an inflection point.

Point of Inflection (POI) To find POIs, find all points that f’’=0 or f’’ DNE and make a number line to test values

Example 3 Determine POI, and where CU and CD

Example 3 cont. Determine POI, and where CU and CD

Second derivative test for local extrema If f’(c)=0 and f’’(c)<0, then f has a local max at x=c If f’(c)=0 and f’’(c)>0, then f has a local min at x=c.

Example 4 Find the coordinates of the relative extrema for f(x) = -3x 5 + 5x 3. Justify using the 2 nd derivative test. Sketch the graph.

Example 5