Concavity and Points of Inflection

3.1 Concavity and Points of Inflection

Concavity

Recall that if the derivative of a function is positive in an interval, then the function is increasing on that interval, and if the derivative is negative, the function is decreasing (see Chapter 2).

Since \(f^{\prime \prime }(x)\) is the derivative of \(f'(x)\), which is equal to the slope of the tangent line, if \(f^{\prime \prime }(x)>0\), the slope is increasing. This means that moving from left to right, the tangent line turns counterclockwise (Figure 3.1.1a) and the curve bends upward. In this case, we say the function (or its graph) is concave up (the concave side of a curve is its hollow side).

Similarly if \(f^{\prime \prime }(x)\) is negative, the slope is decreasing, the tangent turns clockwise (Figure 3.1.1b), and the curve bends downward. In this case, we say the function (or its graph) is concave down.

PIC

PIC

(a) Increasing slope (increasing \(f’\))

(b) Decreasing slope (decreasing \(f’\))

Figure 3.1.1:

Definition 3. A function \(f\) is concave up if \(f’\) is increasing and is concave down if \(f’\) is decreasing.

Theorem 4. If \(f^{\prime \prime }(x)>0\) on an interval \(I\), then \(f\) is concave up on that interval and if \(f^{\prime \prime }(x)<0\), it is concave down.