Understanding Concavity

-2-x+

x coordinate of point P

-5

calculate derivatives?

first derivative at P is -11 (slope of red line)

second derivative at P is 2

A differentiable function

on some interval

is said to be concave up if

is increasing and concave down if

is decreasing. If

is constant, then the function has no concavity. Points where a function changes concavity are called inflection points.

The red line is the tangent to the curve at

and the dashed blue line is the tangent to the curve a little to the right of

Details

To visualize the idea of concavity using the first derivative, consider the tangent line at a point. Recall that the slope of the tangent line is precisely the derivative. As you move along an interval, if the slope of the line is increasing, then

is increasing and so the function is concave up. Similarly, if the slope of the line is decreasing, then

is decreasing and so the function is concave down.

In this Demonstration, the tangent line at

P=(x,f(x))

is drawn in red. The tangent line at

(x+.15,f(x+.15))

is denoted by a dashed blue line. If

never changes sign twice in an interval.15 units wide or smaller, as is the case in all the examples considered by this Demonstration, then whenever the blue line has a larger slope than the red line, the derivative is increasing from

x+.15

and the function is concave up on that interval. Likewise, whenever the blue line has a smaller slope than the red line, the derivative is decreasing from

x+.15

and the function is concave down on that interval.

In practice, we use the second derivative test to check concavity. The second derivative test says that a function

is concave up when

f''>0

and concave down when

f''<0.

This follows directly from the definition as the

is concave up when

is increasing and

is increasing when its derivative

f''