concaveupward

Concave upward, or concave up, describes the curvature of a real-valued function on an interval. A function f is concave upward on I if its second derivative exists on I and f''(x) > 0 for all x in I. Equivalently, the first derivative f' is increasing on I. Geometrically, the graph resembles a cup, opening upward, and tangent lines lie below the graph. The slope of the graph increases as x increases.

If f''(x) ≥ 0 with equality at isolated points, the function is still concave up on those subintervals; if f'' changes sign, the concavity changes, and inflection points occur where f''(x) = 0 or is undefined and the concavity switches.

Examples: f(x) = x^2 is concave upward for all x, since f''(x) = 2 > 0. The function f(x) = e^x is concave upward since f''(x) = e^x > 0. By contrast, f(x) = sqrt(x) on (0, ∞) is concave downward because f''(x) = -1/(4 x^(3/2)) < 0.

Notes: The concept can be extended to functions that are not twice differentiable, using the definition via secant lines: a function is concave up on an interval if the line segment between any two points on the graph lies above or on the graph. In applications, concavity informs optimization and curve sketching, as concave up regions indicate rising slopes and local minima where f' changes from negative to positive.