# Prove that as a function of x, y never decreases

• Poirot
In summary, the conversation discusses the differentiation of the function y= A(x-sin(x)) with A as a constant and determining when the function is decreasing. The conclusion is that the function is only decreasing if A is greater than zero, as shown through a proof using the intermediate value theorem.
Poirot

## Homework Statement

y= A(x-sin(x)) with A as a constant.

## Homework Equations

dy/dx = A(1-cos(x)) ??

## The Attempt at a Solution

If I am thinking about this correctly, one can just differentiate the function as I have, and argue that when the gradient (dy/dx) is less than zero, the function is decreasing. So from this:

A(1-cos(x))<0
so 1-cos(x)<0

cos(x)>1
which never happens for any x, including negative x since cos is an even function.

Is this the right way of doing this? I'm very rusty on proofs etc.

That works, and you can prove it with the intermediate value theorem.

Thank you very much indeed!

Poirot said:

## Homework Statement

y= A(x-sin(x)) with A as a constant.

## Homework Equations

dy/dx = A(1-cos(x)) ??

## The Attempt at a Solution

If I am thinking about this correctly, one can just differentiate the function as I have, and argue that when the gradient (dy/dx) is less than zero, the function is decreasing. So from this:

A(1-cos(x))<0
so 1-cos(x)<0

cos(x)>1
which never happens for any x, including negative x since cos is an even function.

Is this the right way of doing this? I'm very rusty on proofs etc.

Your conclusion is correct only if ##A >0##, which you did not state.

Yeah sorry A is greater than zero, It's a combination of physical constants found earlier in the question.

## 1. What does it mean for a function to never decrease?

A function that never decreases means that the output value (y) is always equal to or greater than the previous output value for any input value (x). In other words, as the input value increases, the output value either stays the same or increases.

## 2. How do you prove that a function never decreases?

To prove that a function never decreases, you can use mathematical techniques such as taking the derivative of the function and showing that it is always positive or using the mean value theorem to show that the function is always increasing. Another way is to graph the function and visually demonstrate that it never decreases.

## 3. Can a function never decrease on a specific interval but decrease on another interval?

Yes, a function can never decrease on a specific interval but decrease on another interval. This means that for some values of x in that interval, the output value will either stay the same or increase, while for other values of x, the output value will decrease.

## 4. Are there any special types of functions that always never decrease?

Yes, there are special types of functions that always never decrease, such as linear functions with a positive slope and exponential functions with a positive base. These types of functions will always increase or stay the same as the input value increases.

## 5. How is a function that never decreases different from a function that is always increasing?

A function that never decreases and a function that is always increasing are similar in that the output value will either stay the same or increase as the input value increases. However, a function that never decreases can have intervals where the output value stays the same, while a function that is always increasing will always have an increasing output value.

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