There was a question posted that read (as I recall) demonstrate that f(x) = x + cos(x) is an increasing function. All of the answers given at the time I looked, used calculus and the derivative to demonstrate that the derivative was positive and therefore the function was increasing. I have no problem with that, but in looking at the function, my first thought was more intuitive and went like this.

If we look at x as it moves from left to right, it's value is always increasing. The cos(x) is always a postive value which will vary with x to values between 0 and 1 and being added to x, which is increasing, will always yeild a larger value. The rate at which the function will increase will vary but the function will always increase in value.

My question, would this explanation on its own be sufficient or would I, as I most likely would have, need to then look for the more mathematical proof/reasoning and bring in the derivative?

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I think that the proper way to prove that a function f is increasing is to show that the first derivative of f is greater than zero. This is probably not the only way, but in my opinion it is definitely the most straightforward approach. I think your first thought is intuitive, and is a good first step when starting the proof. It says "okay, I know intuitively that this function is increasing, so now I can start looking for a more formal way to prove this." If your intuition told you it wasn't, then in that case a good next step would be to look for a counterexample showing that it is not increasing. I think this is a good strategy to take in general with proofs.

The original question posted was not meant to be an exercise for students to develop understanding of some specific concepts of mathematics and improve skills in using these concepts to other problems. The purpose of the question was not to uncover some fact unknown so far or to prove some thing that has not been proved so far. It was not even to prove this to an individual doubting the validity of facts stated. Thus though there can be many different ways to establish the facts required to be proved by the question, the appropriate and sufficient method of proving will depend on the kind of mathematical topic that it relates to. There is no absolute right way of solving the given problem, proving the facts asserted in the question.

We go by calculus sometimes. It is a popular way for some typical situation. We go by an induction method for some situation. We may assume hypothesis to be true and proceed till we arrive at a conrtradiction and make logical contradiction of the hypothesis. We always go by a certain procedure till there is an alternative more rigorous in approach to deel with a (mathematical) problem. A lot of reasoning and logic is there behind all these different procedures and also behind the development of calculus.

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