Convex and Concave Curves

Introduction to Convex and Concave Curves

• Convex curves, or convex functions are curves that curve upwards. Graphically, if a curve is above the line segment connecting any two points on it, the curve is said to be convex.
• Conversely, concave functions curve downwards. A function is classified as concave if it lies beneath the line segment connecting any two points on it.

Determining Convexity and Concavity using Differentiation

• The first derivative of a function reveals information about the function’s increasing or decreasing behaviour. However, the second derivative provides data concerning the shape of the curve, specifically its concavity.
• If the second derivative, denoted as f’‘(x) or d²y/dx², of a function is positive at a particular interval, then the function is convex on that interval.
• If the second derivative of a function is negative at a certain interval, the function is concave during that interval.
• The point at which a curve changes from being concave to convex or vice versa is known as the point of inflexion.

Points of Inflection

• Points of inflexion are points where the curve changes its shape or curvature. The second derivative equals to zero at these points.
• If the second derivative changes sign at the point where it is zero, then the curve has a point of inflexion at that point.

Application

• Convex and concave functions have great importance in various fields such as economics, where they are applied to analyse cost functions, and engineering for designing lenses or mirrors.
• Understanding convexity and concavity is also crucial for optimisation problems, as global and local extrema can only exist in intervals where the function is either entirely concave or convex.