# R twice continuously differentiable function proof

• irresistible
In summary: This conversation is discussing the proof of the given inequality for a twice continuously differentiable function satisfying certain conditions. The proof utilizes the Mean Value Theorem and properties of concave down curves.
irresistible
r twice continuously differentiable function proof...

## Homework Statement

Help
if f:[a,b] $$\rightarrow$$ R is twice continuously differentiable, and f(x)$$\geq$$ 0 for all x in [a.b]
and f ''(x) $$\leq$$ 0 for all x in [a,b]
prove that

1/2 (f(a) + f(b)) (b-a) $$\leq$$ $$\int$$ f(x)dx $$\leq$$(b-a) f(a+b/2)

(integral going from a to b)

## Homework Equations

-It's given that f is twice continuously differentiable.

## The Attempt at a Solution

By looking at the inequality I'm assuming that I have to use the mean value theorem somewhere..
But how?

Here's a hint. The derivative condition tells you that f(x) is concave down on the interval [a,b]. Draw a graph. Draw the secant line connecting x=a and x=b. That's below the curve, right? Now draw the tangent line at x=(a+b)/2. That's above the curve, right? Your two bounds are related to the integrals of those two linear functions. You'll need the MVT to prove the aboveness and belowness parts, if you don't already have such theorems.

the inequality makes sense now that you explained,
I still have trouble proving it,

Where do I use the fact that f is TWICE differentiable?Does that make a difference?

You need twice differentiable to show some properties of your curve related to it being concave down. Like those I pointed out yesterday. You haven't done anything on this problem yet. I'd suggest you get started. Don't PM me about problems, ok?

## 1. What is a "R twice continuously differentiable function"?

A "R twice continuously differentiable function" is a function that has two continuous derivatives. This means that the function is smooth and has no abrupt changes in its slope.

## 2. How do you prove that a function is "R twice continuously differentiable"?

To prove that a function is "R twice continuously differentiable", you must show that it has two continuous derivatives, which means that the limit of the function's derivatives exists and is continuous at all points in its domain. This can be done using the definition of a derivative and the rules of differentiation.

## 3. What is the significance of a function being "R twice continuously differentiable"?

A function being "R twice continuously differentiable" is significant because it means that the function is very smooth and has no abrupt changes in its slope. This makes it easier to analyze and study, as well as use in various mathematical and scientific applications.

## 4. Can a function be "R twice continuously differentiable" but not be continuous?

No, a function cannot be "R twice continuously differentiable" if it is not continuous. This is because in order for a function to have derivatives, it must be continuous at all points in its domain. Therefore, a function that is not continuous cannot have continuous derivatives and cannot be "R twice continuously differentiable".

## 5. How is a "R twice continuously differentiable function" different from a "twice differentiable function"?

A "R twice continuously differentiable function" is different from a "twice differentiable function" in that a "R twice continuously differentiable function" has two continuous derivatives, while a "twice differentiable function" may have two derivatives but they may not be continuous. This means that a "R twice continuously differentiable function" is smoother and has no abrupt changes in its slope, making it easier to analyze and use in mathematical and scientific applications.

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