Concavity and Point of Inflection of Graphs
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.
Example 1: Concavity Up
Let us consider the graph below. 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.
Example 3
Determine the values of the leading coefficient \( a \) for which the graph of function \( f(x) = a x^2 + b x + c \) is concave up or down.Solution to Example 3
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 in the interval \((-8 , + 8)\). 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 \((-8 , + 8)\) since \( f''(x) = 2a \) is negative.
The graphs of two quadratic functions are shown below: \( y = 2x^2 - 2x - 1 \) whose graph is concave up because its leading coefficient (\( a = 2 \)) is positive and \( y = -x^2 + 3x + 1 \) whose graph is concave down because its leading coefficient (\( a = -1 \)) is negative.
Example 4
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 to Example 4
Let us find the first two derivatives of function \( f \).a)
\( f'(x) = 4x^3 - 6x^2 + 1 \)
\( f''(x) = 12x^2 - 12x \)
Find the zeros \( f''(x) \).
\( 12x^2 - 12x = 0 \)
\( 12x(x - 1) = 0 \)
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 of the three intervals and find the sign of \( f'' \)
We now use the table of sign and the theorem above to conclude that
in the interval \((-8 , 0)\); \( f'' \) is positive and therefore the graph of \( f \) is concave up
in the interval \( (0 , 1) \); \( f'' \) is negative and therefore the graph of \( f \) is concave down
in the interval \( (1 , +8) \); \( f'' \) is positive and therefore the graph of \( f \) is concave up
The second derivative \( f'' \) changes sign at \( x = 0 \) and \( x = 1 \) and therefore the graph of \( f \) has two inflection point: \( (0 , f(0)) \) and \( (1 , f(1)) \)
b)
The graph of \( f \) (blue) and \( f'' \) (red) are shown below. It can easily be seen that whenever \( f'' \) is negative (its graph is below the x-axis), the graph of \( f \) is concave down and whenever \( f'' \) is positive (its graph is above the x-axis) the graph of \( f \) is concave up.
Point \( (0,0) \) is a point of inflection where the concavity changes from up to down as \( x \) increases (from left to right) and point \( (1,0) \) is also a point of inflection where the concavity changes from down to up as \( x \) increases (from left to right).
Example 5
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 to Example 5
According to the graph of \( f'' \), the sign of \( f'' \) is given over the following intervalsa)
On the interval \((-\infty , 2)\), the graph of \( f'' \) is below the x-axis and therefore \( f'' \) is negative, hence \( f \) is concave down on the interval \((-\infty , 2)\)
On the interval \( (2 , +\infty) \), the graph of \( f'' \) is above the x-axis and therefore \( f'' \) is positive, hence \( f \) is concave up on the interval \( (2 , +\infty) \)
At \( x = 2 \), the sign of \( f'' \) changes and therefore \( x = 2 \) is a point of inflection.
Example 6
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 to Example 6
We use the graph of the first derivative \( f' \) to find the sign of the second derivative and deduce the concavity of the graph of \( f \)a)
On the interval \((-\infty , -2)\), \( f' \) decreases and therefore \( f'' \) is negative; the graph of \( f \) is concave down
On the interval \( (-2 , -1) \), \( f' \) increases and therefore \( f'' \) is positive; the graph of \( f \) is concave up
On the interval \( (-1 , 1) \), \( f' \) decreases and therefore \( f'' \) is negative; the graph of \( f \) is concave down
On the interval \( (1 , +\infty) \), \( f' \) increases and therefore \( f'' \) is positive; the graph of \( f \) is concave up
The concavity of the graph of \( f \) changes at \( x = -2 \), \( x = -1 \) and \( x = 1 \) and therefore these are all point of inflection.
More References and links
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