Glossary

Concave down

Criticality: 3

A function is concave down if its graph opens downward, meaning the slopes of its tangent lines are decreasing. This occurs when the second derivative of the function is negative.

Example:

The function $f(x) = -x^2$ is always concave down, resembling an inverted U-shape, as its second derivative is $f''(x) = -2$ .

Concave up

Criticality: 3

A function is concave up if its graph opens upward, meaning the slopes of its tangent lines are increasing. This occurs when the second derivative of the function is positive.

Example:

The function $f(x) = e^x$ is always concave up because its second derivative, $f''(x) = e^x$ , is always positive.

Concavity

Criticality: 3

Concavity describes the direction a function's graph opens, either upward or downward. It is determined by the rate at which the slopes of the tangent lines are changing.

Example:

The graph of $y=x^4$ exhibits concavity that is always upward, forming a wide U-shape.

Inflection point

Criticality: 3

An inflection point is a point on the graph of a function where the concavity changes (from concave up to concave down or vice versa). At this point, the second derivative is typically zero or undefined, and its sign changes.

Example:

For $f(x) = x^3$ , the origin $(0,0)$ is an inflection point because the graph changes from concave down to concave up at that specific location.

Second derivative

Criticality: 3

The second derivative of a function, denoted as $f''(x)$ or $\frac{d^2y}{dx^2}$, is the derivative of its first derivative. It provides information about the concavity of the original function.

Example:

If a car's position is given by $s(t)$ , its velocity is $s'(t)$ , and its acceleration is the second derivative, $s''(t)$ .