I'm not sure what you mean here. You want to prove that the straight line between (a,f(a)) and (b,f(b)), which is y= (f(b)-f(a))/(b-a) (x- a)+ f(a) lies above the curve y= f(x). That is, that (f(b)-f(a))/(b-a) (x- a)+ f(a)> f(x) for all x between a and b.barksdalemc said:Homework Statement
Let f be differentiable on (a,b). Show that f is convex if and only if for every x,y in (a,b), f(y)-f(x)>= (y-x)f'(x)
The Attempt at a Solution
The mean value theorem says that there exists an x' in (a,b) such that f'(x') is the average rate of change of the functions. So I have the equation for that tangent line. I am stuck there.
Good heaven's no! There are plenty of theorems that are true in one direction but false in the other!barksdalemc said:If I prove one direction, is the proof in the other direction just the logic going the other way? In any case, let's say I want to show it is convex given for every x,y in (a,b), f(y)-f(x)>= (y-x)f'(x)
barksdalemc said:If I prove one direction, is the proof in the other direction just the logic going the other way?
The Mean Value Theorem is a fundamental theorem in calculus that states that for any continuous and differentiable function on a closed interval, there exists at least one point in the interval where the slope of the tangent line is equal to the average rate of change of the function over that interval.
The Mean Value Theorem can be used to prove that a function is convex by showing that the slope of the secant line connecting any two points on the function's graph is always greater than or equal to the slope of the tangent line at any point between those two points. This is because a convex function always lies above its tangent lines.
No, the Mean Value Theorem can only be used to prove the convexity of continuous and differentiable functions. If a function is not continuous and differentiable, then the Mean Value Theorem cannot be applied.
Yes, there are other methods for proving convexity of functions such as using the second derivative test, the convexity criteria, or the Hessian matrix. However, the Mean Value Theorem is a commonly used method because it is relatively simple and intuitive.
Proving convexity of a function is important in many areas of mathematics, economics, and engineering. It helps in identifying the minimum or maximum values of a function, optimizing solutions to problems, and understanding the behavior of functions in various contexts.