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Home / Calculus I / Applications of Derivatives / The Shape of a Graph, Part I
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Section 4.5 : The Shape of a Graph, Part I

14. Given that \(f\left( x \right)\) and \(g\left( x \right)\) are increasing functions. If we define \(h\left( x \right) = f\left( x \right) + g\left( x \right)\) show that \(h\left( x \right)\) is an increasing function.

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Hint : At first glance this problem may seem quite difficult. However, just keep in mind how we have been determining whether functions where increasing to this point and that should suggest a first step.
Start Solution

To this point we’ve always used the derivative to determine if a function was increasing so let’s do that here as well.

Note that this may not seem all that useful because we don’t actually know what any of the functions are. However, just because we don’t know what the functions are doesn’t mean that we can’t at least write down a formula for \(h'\left( x \right)\). Here is that formula.

\[h'\left( x \right) = f'\left( x \right) + g'\left( x \right)\]
Hint : We were told that \(f\left( x \right)\) and \(g\left( x \right)\) are increasing functions so what does that tell us about their derivatives?
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We are told that both \(f\left( x \right)\) and \(g\left( x \right)\) are increasing functions so this means that we know that both of their derivatives must be positive. Or,

\[f'\left( x \right) > 0\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}g'\left( x \right) > 0\]
Hint : Given the formula for \(h'\left( x \right)\) we found in Step 1 and what we noticed about the signs of \(f'\left( x \right)\) and \(g'\left( x \right)\) that we noted in Step 2 what can we say about the sign of \(h\left( x \right)\)?
Show Step 3

Okay, we are pretty much done at this point. We know from Step 1 that,

\[h'\left( x \right) = f'\left( x \right) + g'\left( x \right)\]

Also from Step 2 we know that both \(f'\left( x \right)\) and \(g'\left( x \right)\) are positive. So, \(h'\left( x \right)\) is the sum of two positive functions and in turn means that we must have,

\[h'\left( x \right) > 0\]

Therefore we can see that \(h\left( x \right)\) must be an increasing function.

Final Thoughts / Strategy Discussion

Figuring out how to do these kinds of problems can definitely seem quite daunting at times. That is especially true when the statement we are being asked to prove seems to be fairly "obvious" as is the case here. The sum of two increasing functions intuitively should also be increasing. The problem is that we are being asked to actually prove that and not just say "well it makes sense so it should be true".

What we want to discuss here is not the proof of this fact (that is given above after all...). Instead let's take a look at the thought process that went into constructing the proof above.

The first step is to really look at what we are being asked to prove. This means not just reading the statement, but reading the statement and trying to relate what we are being asked to prove to something we already know.

In this case, we’re being asked to show that a general function is increasing given a set of assumptions. By this point we know how to prove that specific functions are increasing. So, let’s start with that.

We know that in order to use Calculus to prove that a function is increasing we need to look at the derivative of the function. We also know that, at least symbolically, we write down the formula for the derivative of the function we are interested in for this problem.

Now for the next step in the thought process. We've got a formula and we know that we need to show that it is positive. At this point we need to think about the assumptions that we were given. Don't forget the assumptions. They were given for a reason and we'll need to use them. What do the assumptions tell us? How can we relate them to what we are being asked to prove?

In this case, we know from the assumptions that the two derivatives were positive.

For this problem this wasn't a particularly difficult step, but for other problems this can be a little tricky.

Finally, we need to put the two previous thought process steps together. This can also be a fairly tricky step. If you haven't had a lot of exposure to "mathematical logic/proofs" it can be daunting to put all the information together. Often times you will need to try various ways of putting the information together before something "clicks" and you can see how to proceed. You may even need to go back to the previous step and see if there is something about the assumptions that you may have missed.

In this case we could see that the derivative was the sum of two other derivatives and from our assumptions we knew that the two individual derivatives had to be positive. We also know from basic Algebra knowledge that the sum of two positive quantities has to be positive and so we are done.

The key part of this whole process is that you will have to persevere. Try not to get discouraged and if something doesn't work out move on and try something else. Also, do not get so wrapped up in the process that you don't take breaks occasionally. If you keep running into road blocks then step away for a while and come back at a later point. Sometimes that is all it takes to get a fresh idea.

Another thing that students often initially have difficulty with is trying to mathematically write this kind of thing down. In your mind you may have been able to see all the "logic" involved in the proof, but just couldn't see how to put it all together and write it down. If you are having that problem the best thing to do is just start writing things down.

For instance, you know you need the derivative of the given function so write that down. If you don't have a specific function to differentiate can you at least symbolically write down the derivative as we did here?

Once you have that written down look at the pieces and start writing down what you know about them. Actually write down what you know (i.e. things like \(f'\left( x \right) > 0\)). This seems silly at times, but it really can help with the process.

Once you have everything written down you might be able to see how to string everything together with words/explanations to prove what you want to prove.