Understanding Convex and Concave Shapes and Functions in Mathematics

Convex and concave

Convex and concave are terms used in mathematics to describe the curvature of a shape or a function

Convex and concave are terms used in mathematics to describe the curvature of a shape or a function.

Convex:
A shape or a function is said to be convex if, for any two points within the shape or on the graph of the function, the line connecting those two points lies completely inside the shape or above the graph of the function. In other words, a shape is convex if it does not have any indentations or “dents” in it. Similarly, a function is convex if its graph is always “above” any line segment connecting two points on the graph. Convex shapes have a continuous outward curvature and do not form any inward curves or concave sections.

Examples of convex shapes include circles, squares, and regular polygons. In terms of functions, any function where the second derivative is non-negative throughout its domain is considered convex.

Concave:
Conversely, a shape or a function is said to be concave if, for any two points within the shape or on the graph of the function, the line connecting those two points lies completely below the graph of the function or outside the shape. A concave shape or function forms inward curves or “dents”. In other words, if you were to place a straight edge on a concave shape, it would not touch all points of the shape. Similarly, if you were to draw a line segment connecting two points on the graph of a concave function, it would not lie completely above the graph.

Examples of concave shapes include crescent moons or the inside of a bowl. In terms of functions, any function where the second derivative is non-positive throughout its domain is considered concave.

It’s important to note that a shape or function can also be neither convex nor concave. In such cases, we typically refer to them as a non-convex or non-concave shape/function.

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