Concavity of a Function
Let \( y=f(x) \) be a function defined and differentiable on an interval \( I \). At every interior point \(x_0 \), the graph of the function has a tangent line, which we denote by \(y=t(x) \). By comparing the graph with its tangent line, we can determine whether the function is concave upward or concave downward.
- Concave upward
A function is said to be concave upward at the point \(x_0 \) if there is an open neighborhood \(I_{x_0} \) of \(x_0 \) such that, for every point in the neighborhood other than \(x_0 \), the graph lies above the tangent line. \[ f(x)>t(x) \qquad \forall x \in I_{x_0}, \ x\neq x_0 \] In this situation, the tangent line lies below the graph of the function. A function that is concave upward on an interval is also called convex on that interval.
- Concave downward
A function is said to be concave downward at the point \(x_0 \) if there is an open neighborhood \(I_{x_0} \) of \(x_0 \) such that, for every point in the neighborhood other than \(x_0 \), the graph lies below the tangent line. \[ f(x)<t(x) \qquad \forall x \in I_{x_0}, \ x\neq x_0 \] In this situation, the tangent line lies above the graph of the function. A function that is concave downward on an interval is simply called concave on that interval.
In simple terms, concavity describes how the graph of a function bends. A graph that bends upward is concave upward, while a graph that bends downward is concave downward.
From a geometric point of view, the type of concavity can be determined by comparing the graph of the function with its tangent line.
A Practical Example
Consider the function
\[ f(x)=x^2 \]
At the point \( x_0=0 \), the tangent line coincides with the \( x \)-axis, so its equation is
\[ y=0 \]
In a neighborhood of the origin,
\[ x^2>0 \qquad (x\neq0) \]
Since the graph of the parabola always lies above its tangent line, the function is concave upward.
Example 2
Now consider the function
\[ f(x)=-x^2 \]
At the point \( x_0=0 \), the tangent line is again
\[ y=0 \]
For every nonzero point in a neighborhood of the origin,
\[ -x^2<0 \]
This time, the graph lies below the tangent line, so the function is concave downward.
How to Determine Concavity
The geometric idea is straightforward. Simply compare the graph with its tangent line.
- If the graph lies above the tangent line, the function is concave upward.
- If the graph lies below the tangent line, the function is concave downward.
This geometric interpretation leads directly to the second derivative test, which allows us to determine concavity without comparing the graph and the tangent line point by point.
In practice, the geometric approach is rarely used because it would require analyzing the relative position of the graph and its tangent line at every point.
Instead, concavity is almost always determined analytically by studying the sign of the second derivative.
Relationship Between Concavity and the Second Derivative
If a function is twice differentiable, the sign of its second derivative immediately reveals the concavity of the graph over a given interval \( I \).
- If \( f''(x)>0 \) for every \(x\) in \(I\), the function is concave upward on \(I\).
- If \( f''(x)<0 \) for every \(x\) in \(I\), the function is concave downward on \(I\).
For most functions, analyzing the sign of the second derivative is the quickest and most reliable way to determine concavity and locate any points of inflection.
Example
Consider the function
\[ f(x)=x^2 \]
Without looking at the graph, first compute the first derivative:
\[ f'(x)=2x \]
Then compute the second derivative:
\[ f''(x)=2 \]
The second derivative is constant and positive for every value of \(x \):
\[ f''(x)=2>0 \qquad \forall x \in \mathbb{R} \]
Since the second derivative is always positive, the function is concave upward throughout its domain.
Note. This agrees with the familiar shape of the parabola \(y=x^2 \), which opens upward. At every point on the curve, the tangent line lies below the parabola.
Example 2
Now consider the function
\[ f(x)=-x^2 \]
First compute the first derivative:
\[ f'(x)=-2x \]
Then compute the second derivative:
\[ f''(x)=-2 \]
The second derivative is constant and negative for every value of \(x \):
\[ f''(x)=-2<0 \qquad \forall x \in \mathbb{R} \]
Since the second derivative is always negative, the function is concave downward throughout its domain.
Note. This is consistent with the graph of the parabola \( y=-x^2 \), which opens downward. At every point on the curve, the tangent line lies above the parabola.
In general, determining the concavity of a function is as simple as computing its second derivative and analyzing its sign over the different intervals of the domain.
And so on.
