concave up

A function is concave up on an interval if its second derivative is positive, indicating that the first derivative is increasing and the slope of the tangent line to the function is becoming steeper. This concave up property implies that the tangent line to the function lies below the curve throughout the interval. Conversely, a function is concave down when its second derivative is negative, meaning the first derivative is decreasing and the tangent line lies above the curve.

Applications

Concave up describes a function's behavior where its graph lies above all its tangent lines, and the derivative of the function is increasing.

Notes & references