Is f convex on (a,b) if and only if f(y)-f(x)>(y-x)f'(x) for all x, y in (a,b)?

[barksdalemc]

Can someone tell me if my logic in answering the following questions is ok:

1. Suppose that f(x) and g(x) are convex on (a,b). Show that the functions h(x)=max[f(x), g(x)] is also convex on (a,b).

-I said that since f and g are convex their second derivatives are not equal to zero in (a,b) and then said since h is the max of f or g, that h also has a second derivative in (a,b) not equal to zero implying it is also convex.

2. Let f be differentiable on (a,b). Show that if f is convex if and only if for all x, y in (a,b): f(y)-f(x)>(y-x)f'(x)

-I'm at a loss on this one.

 

[StatusX]

What is your definition of convex? I guarantee it is not equivalent to "having a second derivative not equal to zero." Not only do convex function need not have even first derivatives, let alone second ones, but those that do certainly can have second derivatives equal to zero, such as any line.

 

[barksdalemc]

ok. the definition states that a curve is convex in a ndb about some x if every point on the curve in the nbd is either above or below the tangent line at x.

 

[lotrgreengrapes7926]

2. Let f be differentiable on (a,b). Show that if f is convex if and only if for all x, y in (a,b): f(y)-f(x)>(y-x)f'(x)

Rearrange: f(y)$$\ge$$f(x)+(y-x)f'(x).
Draw a diagram: Two points (x,f(x)) and (y,f(y)), the horizontal distance between (y-x) and slope at (x,f(x)) is f'(x).

 

[barksdalemc]

Thanks, but it seems like rewording of the definition of convexity. Is that ok?

 

[lotrgreengrapes7926]

Convex is when the curve is never below any tangent line (it can be above or contained in the tangent).