I’m teaching a Calculus course this year and came across an interesting problem the other day. We were working from the Briggs & Cochran Calculus book and discussing the quotient rule. Problem #40 on page 127 says given the function:

a) Find the values of for which the slope of the curve is zero.

b) Does the graph of have a slope of at any point?

OK – so we need to apply the quotient rule and find

OR

OK – so part (a) is easy enough. The derivative is equal to zero when . But what about part (b) ? We can graph the derivative and see that it’s always less than , but what about algebraically?

The problem can be posed in a number of different ways. I chose to set it up as:

so

and

Again, we can graph this and SEE clearly that the graph of is definitely above the graph of , but I like to know what’s happening algebraically. In fact, I used to give my College Algebra students algebraic inequalities as an introductory project, so I really wanted to figure this out.

Of course, this came up in the middle of class, so we looked at the graph and I began to consider the algebraic reasoning, but didn’t get far, so I told the class that I would think about it while they had their quiz – but still to no avail. I worked on it during my free time yesterday and made some progress, then when I came in this morning it all fell together:

OK, done there.

Again – ok, done. What was the sticking point for me yesterday were the positive values of . So:

So,

Also,

So,

So now we know that and can work our way back to

For some reason, I find this beautiful!