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
Another interactive tutorial, using an applet, on the concavity of graphs quadratic functions is included in this site.
More on applications of differentiation
applications of differentiation