Question

Suppose we have a function f : r → r that is convex and differentiable, with r being a subset of its domain. we want to demonstrate that the function defined as 1 z x g ( x ) = f ( t ) dt for dom ( g ) = r x 0 is a convex function.

A) g(x) is not necessarily convex.
B) g(x) is convex for all x in R.
C) g(x) is convex only when f(x) is a linear function.
D) g(x) is concave for all x in R.

Answer 1

The function g(x) defined by the integral of a convex, differentiable function f(t) preserves the convexity property, thus g(x) is convex for all x in R.

Step-by-step explanation:

Given a convex and differentiable function f defined on a subset of its domain r, we are asked to investigate the convexity of the function g(x) defined by the integral g(x) = ∫ f(t) dt from a constant x_0 to x where x is in r.

To prove that g(x) is convex, we can consider the property of convex functions which states that if a function f is convex over an interval, then the line segment between any two points on the graph of f lies above or on the graph. Since integration, in this case, is the accumulation of the area under f(t), it preserves this property, implying that the integral g(x) will also satisfy the definition of a convex function for all x in R.

Therefore, option B is correct: g(x) is convex for all x in R.