The definition of the concavity of a graph is introduced along with inflection points. Examples, with detailed solutions, are used to clarify the concept of concavity using first and second derivatives.
Visual Examples of Concavity
Example 1: Concavity Up
Note that the slope of the tangent line (first derivative) increases. The graph in the figure below is called concave up.
Figure 1
Example 2: Concavity Down
The slope of the tangent line (first derivative) decreases in the graph below. We call the graph below concave down.
Figure 2
Definition of Concavity
Let \( f' \) be the first derivative of function \( f \) that is differentiable on a given interval \( I \). The graph of \( f \) is:
- (i) concave up on the interval \( I \), if \( f' \) is increasing on \( I \), or
- (ii) concave down on the interval \( I \), if \( f' \) is decreasing on \( I \).
The sign of the second derivative informs us when \( f' \) is increasing or decreasing.
Theorem
Let \( f'' \) be the second derivative of function \( f \) on a given interval \( I \). The graph of \( f \) is:
- (i) concave up on \( I \) if \( f''(x) > 0 \) on the interval \( I \).
- (ii) concave down on \( I \) if \( f''(x) < 0 \) on the interval \( I \).
Definition of Point of Inflection
A point \( P \) on the graph of \( y = f(x) \) is a point of inflection if \( f \) is continuous at \( P \) and the concavity of the graph changes at \( P \). In view of the above theorem, there is a point of inflection whenever the second derivative changes sign.
Worked Examples
Example 3: Quadratic Functions
Problem: Determine the values of the leading coefficient \( a \) for which the graph of function \( f(x) = ax^2 + bx + c \) is concave up or down.
Solution:
We first find the first and second derivatives of function \( f \):
\[ f'(x) = 2ax + b \] \[ f''(x) = 2a \]We now study the sign of \( f''(x) \), which is equal to \( 2a \):
- If \( a \) is positive, \( f''(x) \) is positive on the interval \((-\infty, +\infty)\). According to the theorem above, the graph of \( f \) will be concave up for positive values of \( a \).
- If \( a \) is negative, the graph of \( f \) will be concave down on the interval \((-\infty, +\infty)\) since \( f''(x) = 2a \) is negative.
The graphs of two quadratic functions are shown below: \( y = 2x^2 - 2x - 1 \) (concave up because \( a = 2 > 0 \)) and \( y = -x^2 + 3x + 1 \) (concave down because \( a = -1 < 0 \)).
Example 4: Polynomial Function Concavity and Inflection Points
Problem:
a) Find the intervals on which the graph of \( f(x) = x^4 - 2x^3 + x \) is concave up, concave down, and the point(s) of inflection if any.
b) Use a graphing calculator to graph \( f \) and confirm your answers to part a).
Solution:
Let us find the first two derivatives of function \( f \):
\[ f'(x) = 4x^3 - 6x^2 + 1 \] \[ f''(x) = 12x^2 - 12x \]Find the zeros of \( f''(x) \):
\[ 12x^2 - 12x = 0 \] \[ 12x(x - 1) = 0 \]Two zeros: \( x = 0 \) and \( x = 1 \).
Study sign of \( f'' \):
The two zeros split the set of real numbers into three intervals. Select a value for \( x \) in each interval:
Using the table of signs and the theorem above:
- On the interval \((-\infty, 0)\); \( f'' \) is positive \(\implies\) graph of \( f \) is concave up.
- On the interval \((0, 1)\); \( f'' \) is negative \(\implies\) graph of \( f \) is concave down.
- On the interval \((1, +\infty)\); \( f'' \) is positive \(\implies\) graph of \( f \) is concave up.
The second derivative \( f'' \) changes sign at \( x = 0 \) and \( x = 1 \), therefore the graph of \( f \) has two inflection points: \( (0, f(0)) \) and \( (1, f(1)) \).
b) Graphical Confirmation:
The graph of \( f \) (blue) and \( f'' \) (red) are shown below. Whenever \( f'' \) is negative, \( f \) is concave down; whenever \( f'' \) is positive, \( f \) is concave up.
Example 5: Analyzing from the Second Derivative Graph
Problem: The graph of the second derivative \( f'' \) of function \( f \) is shown below. Find the intervals where \( f \) is concave up, concave down, and the point(s) of inflection if any.
Solution:
- On the interval \((-\infty, 2)\), the graph of \( f'' \) is below the x-axis (\( f'' < 0 \)), hence \( f \) is concave down.
- On the interval \((2, +\infty)\), the graph of \( f'' \) is above the x-axis (\( f'' > 0 \)), hence \( f \) is concave up.
- At \( x = 2 \), the sign of \( f'' \) changes, and therefore \( x = 2 \) is a point of inflection.
Example 6: Analyzing from the First Derivative Graph
Problem: The graph of the first derivative \( f' \) of function \( f \) is shown below. Find the intervals where the graph of \( f \) is concave up, concave down, and the point(s) of inflection if any.
Solution:
We use the graph of the first derivative \( f' \) to find the sign of the second derivative (\( f'' = (f')' \)) and deduce the concavity:
- On the interval \((-\infty, -2)\), \( f' \) decreases (\( f'' < 0 \)) \(\implies\) graph of \( f \) is concave down.
- On the interval \((-2, -1)\), \( f' \) increases (\( f'' > 0 \)) \(\implies\) graph of \( f \) is concave up.
- On the interval \((-1, 1)\), \( f' \) decreases (\( f'' < 0 \)) \(\implies\) graph of \( f \) is concave down.
- On the interval \((1, +\infty)\), \( f' \) increases (\( f'' > 0 \)) \(\implies\) graph of \( f \) is concave up.
The concavity of the graph of \( f \) changes at \( x = -2 \), \( x = -1 \), and \( x = 1 \); therefore, these are all points of inflection.