According to the second-order condition for convexity, what must be true for a function to be convex?
Answer (1)
To determine whether a function is convex, we can use a condition based on the function's second derivative, known as the second-order condition for convexity. This is particularly useful for functions that are twice differentiable.
Definition of Convexity : A function f ( x ) is said to be convex on an interval if, for any two points x 1 and x 2 in the interval and any λ such that 0 ≤ λ ≤ 1 , the following inequality holds: f ( λ x 1 + ( 1 − λ ) x 2 ) ≤ λ f ( x 1 ) + ( 1 − λ ) f ( x 2 ) This means that the line segment joining any two points on the graph of the function lies above or on the graph.
Second-Order Condition : For a twice-differentiable function f ( x ) , we say that it is convex if its second derivative f ′′ ( x ) is non-negative for all x in the interval. Mathematically, this is expressed as: f ′′ ( x ) ≥ 0 for all x in the interval This condition implies that the slope of the tangent to the function increases with x , or at the very least, remains constant. When the second derivative is strictly greater than zero 0"> f ′′ ( x ) > 0 , the function is strictly convex.
Understanding the Condition :
If the second derivative is positive, the function's curve opens upward, indicating it is convex.
If the second derivative equals zero, the function is linear over that interval, which can still be considered convex by definition.
Examples : Consider the quadratic function f ( x ) = a x 2 + b x + c :
For this function, f ′′ ( x ) = 2 a . The function is convex if a ≥ 0 .
By using the second-order condition for convexity, you can easily test many twice differentiable functions for convexity, which is a common requirement in optimization problems and various mathematical applications.
