PolyU AMA1120 Lec 2 Concavity

Definition

Let $f$ be differentiable on an open interval $I$:

• The graph of $f$ is concave upward if $f'$ is increasing on $I$.
• The graph of $f$ is concave downward if $f'$ is decreasing on $I$.

Test

Let $f$ be a function whose second derivative exists on an open interval $I$:

• If $f''(x) > 0$ for all $x \in I$, then the graph of $f$ is concave upward on $I$.
• If $f''(x) < 0$ for all $x \in I$, then the graph of $f$ is concave downward on $I$.

Separation

The following types of points separate the intervals on which $f$ is concave upward and concave downward:

• Points where $f''(x) = 0$ or $f''(x)$ is undefined
• Points where $f(x)$ is undefined

Inflection

A point on the graph of $f$ where concavity changes is called a point of inflection (or inflection point).

We have:

• If $(c, f(c))$ is a point of inflection of the graph of $f$, then $f''(c) = 0$ or $f''(c)$ is undefined.
• A point where $f''(x) = 0$ or $f''(x)$ is undefined is not necessarily a point of inflection.

Second Derivative Test

If $f'(c) = 0$ and $f''(x)$ exists on an open interval containing $c$, then:

• If $f''(c) > 0$, then $f$ has a relative minimum at $c$.
• If $f''(c) < 0$, then $f$ has a relative maximum at $c$.
• If $f''(c) = 0$, then the test is inconclusive.