Second derivative positive implikes midpoint convex

In summary, the conversation discusses using Taylor's theorem with a specific value of h to prove midpoint convexity for a twice differentiable function with a positive second derivative. The problem arises when trying to show that the sum of terms involving the first and second derivative is nonpositive. The suggestion is to use a mean value theorem approach instead of Taylor series. The speaker eventually figures it out.
  • #1
resolvent1
24
0
I've been trying to use Taylor's theorem with h = (y-x)/2 to show that a twice differentiable function for which the second derivative is positive is midpoint convex (ie, f( (1/2)*(x+y) ) \leq (1/2) * (f(x)+f(y)) ). (It's not a homework problem.) The problem I end up with this is that I'm not sure how to show that the sum of the terms involving the first and second derivative is nonpositive. How would I go about showing this, or is there a better (non-Taylor) way to do it?
 
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  • #2
Taylor series seems overwhelming for this kind of problem. I would try using a mean value theorem styled argument (post again if you need further guidance)
 
  • #3
THanks, I've got it.
 

1. What does it mean for a function to have a positive second derivative?

A positive second derivative means that the rate of change of the slope of the function is increasing. In other words, the function is getting steeper at a faster rate.

2. How is the second derivative related to the convexity of a function?

The second derivative is a measure of the curvature of a function. A positive second derivative indicates that the function is concave up, or convex, at that point. This means that the function is "smiling" and has a minimum value at that point.

3. What is the significance of the midpoint being convex in terms of the function's behavior?

If a function is midpoint convex, it means that any midpoint between two points on the function's curve will also lie on or above the curve. This property is important because it guarantees that the function will never dip below the line connecting two points, making it a useful tool in optimization problems.

4. How does the second derivative test help in identifying inflection points?

The second derivative test states that if the second derivative changes sign at a point, then that point is an inflection point. This means that the curvature of the function changes from convex to concave (positive to negative second derivative) or vice versa, indicating a change in the direction of the function's curve.

5. Can a function have a positive second derivative without being midpoint convex?

Yes, a function can have a positive second derivative without being midpoint convex. For example, a function with a positive second derivative can have a "frowning" shape, where it curves downward and then back up. In this case, the function would not have the property of midpoint convexity.

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