What defines a convex function?

A convex function is characterized by its bowl-shaped upward contour, which means that any line segment joining two points on the graph of the function lies above or on the graph itself. This property signifies that as you move along the function from left to right, the slope of the tangent (derivative) doesn't decrease; it either remains constant or increases, indicating that the rate of change is non-decreasing.

This upward shape is crucial in optimization problems, especially in linear programming, because it ensures that local minima found within feasible regions are also global minima. The identification of a convex function allows for the application of various mathematical tools and theories which guarantee convergence to an optimal solution easily.

While a bowl-shaped downward function would indicate a concave function, which has different implications for optimization and differs from the characteristics of convexity, the other choices involving linear functions and decreasing returns do not encapsulate the inherent definition of convexity in mathematical terms. Thus, the correct answer accurately captures the essential trait of a convex function.