The concavity of functions may be determined using the sign of the second derivative. For a quadratic function f 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.