The concavity of functions may be determined using the sign of the second derivative. For a quadratic function f is of the form
f(x) = a x 2 + b x + c , with a not equal to 0
The first and second derivatives of are given by
f '(x) = 2 a x + b
f "(x) = 2 a
The sign of f " depends on the sign of coefficient a included in the definition of the quadratic function. Two cases are possible. If a is positive then f " is positive and the graph of f is concave up. If a is negative then the graph of f is concave down. Below are some examples with detailed solutions.
Example 1
What is the concavity of the following quadratic function?
f(x) = (2 - x)(x - 3) + 3
Solution to Example 1
Expand f(x) and rewrite it as follows
f(x) = -x 2 + 5x -3
The leading coefficient a is negative and therefore the graph of is concave down. see figure below.
Example 2
What is the concavity of the following quadratic function?
f(x) = -2(x - 1)(x - 2) + 3 x 2
Solution to Example 2
Expand f(x) as follows
f(x) = x 2 + 6 x - 4
The leading coefficient a is positive and therefore the graph of is concave up. see figure below.
Exercises With Answers
Determine the concavity of each quadratic function.
a) f(x) = 2x 3 + 6 x - 13
b) f(x) = (2 - x)(4 - x)
c) f(x) = -2(x - 3) 2 - 5
d) f(x) = x(x + 3) - 2(x - 3) 2
Answers to Above Exercises
a) concave up
b) concave up
c) concave down
d) concave down
