Let \(f\) be a function and \(I\) an interval contained in the domain of \(f\).

• We say that \(f\) is concave up on \(I\) if for all numbers \(x_1, x_2, x_3 \in I\), if \(x_1 < x_2 < x_3\) then the point \((x_2, f(x_2))\) is below the secant line through the points \((x_1, f(x_1))\) and \((x_3, f(x_3))\).
• We say that \(f\) is concave down on \(I\) if for all numbers \(x_1, x_2, x_3 \in I\), if \(x_1 < x_2 < x_3\) then the point \((x_2, f(x_2))\) is above the secant line through the points \((x_1, f(x_1))\) and \((x_3, f(x_3))\).

Notes:

• Paul's notes has different definition on concavity

Suppose that \(f\) is a function and \(I\) is an interval contained in the domain of \(f\).

• The function \(f\) is concave up on \(I\) if and only if for all numbers \(x_1, x_2, x_3 \in I\), if \(x_1 < x_2 < x_3\) then

  \(\dfrac{f(x_2) - f(x_1)}{x_2 - x_1} < \dfrac{f(x_3)-f(x_2)}{x_3 - x_2}\)

• The function \(f\) is concave down on \(I\) if and only if for all numbers \(x_1, x_2, x_3 \in I\), if \(x_1 < x_2 < x_3\) then

  \(\dfrac{f(x_2) - f(x_1)}{x_2 - x_1} > \dfrac{f(x_3)-f(x_2)}{x_3 - x_2}\)

Suppose \(f\) is continous on an interval \(I\) and twice differentiable on the interior of \(I\); in other words, for all \(x\) in the interior of \(I\), \(f''(x)\) is defined.

• If \(f''(x) > 0\) for all \(x\) in the interior of \(I\), then \(f\) is concave up on \(I\).
• If \(f''(x) < 0\) for all \(x\) in the interior of \(I\), then \(f\) is concave down on \(I\).

A point \((a,f(a))\) on the graph of a function \(f\) is called an inflection point if there is a number \(d> 0\) such that either \(f\) is concave up on \((a-d, a]\) and concave down on \([a, a + d)\), or \(f\) is concave down on \((a-d, a]\) and concave up on \([a, a +d)\)

Suppose that \(f'(a) = 0\)

1. If \(f''(a) > 0\), then \(f\) has a local minimum at \(a\).
2. If \(f''(a) < 0\), then \(f\) has a local maximum at \(a\).