# A Cubic Conundrum

In class, we tackled a Senior Maths Challenge problem that goes something like this:

“Given that $y^3 f(x) = x^3 f(y)$ for all real $x$ and $y$, and that $f(3) \ne 0$, find the value of $\frac{f(20)-f(2)}{f(3)}$.

A: 6; B: 20; C: 216; D: 296; E:2023”

Have a play with it. Spoilers below the line. The interesting bits for me come after the problem is solved.

The question itself is more intimidating than it is difficult. In particular, $27 f(x) = x^3 f(3)$ for all $x$, so we can multiply by 27 and replace all of the $f$ evaluations like this ((This is simply the neatest way I can see; other options are available.)):

$\frac{27 f(20)- 27f(2)}{27f(3)} = \frac{20^3 f(3) - 2^3 f(3)}{27 f(3)}$.

All of the $f(3)$s cancel, leaving $\frac{8000 - 8}{27}$.

A quick sense check says that’s between 100 and 1000, so the answer must be C or D; use whatever arithmetic you like to find that it’s 296. (Under pressure, I’d probably divide by 9 and by 3; alternatively, $27\times 250 = 6750$, so the answer is greater than 250). The answer is D.

### A wry twist

The answer can’t be 216, because 216 is a cube. If the answer were 216 then $20^3 + (-2)^3 = \br{3^3}{6^3}$, which would be a counterexample to Fermat’s Last Theorem.

### But what’s the function?

I also wondered what functions might satisfy the given relationship, which I rewrote as $\frac{f(y)}{f(x)} = \frac{y^3}{x^3}$. It’s fairly obvious that $f(x) = x^3$ satisfies this, but are there others?

I don’t much like the division. I prefer subtraction. I’m going to let $F(x) = \ln(f(x))$, so that $F(y) - F(x) = 3\ln(y) - 3\ln(x)$.

Now let $y = x + h$: $F(x+h) - F(x) = 3\ln(x+h) - 3\ln(x)$

Divide by $h$: $\frac{F(x+h)-F(x)}{h} = 3\frac{ \ln(x+h) - \ln(x) }{h}$

So the derivative of $F(x)$ is equal to the derivative of $3\ln(x)$ – or $\diff{}{x} \ln(f(x)) = \diff{}{x}\ln(x^3)$.

Integrate both sides with respect to $x$: $\ln(f(x)) = \ln(x^3)+ C$

So $f(x) = Ax^3$ for any constant $A$ ((Except for 0, which is disallowed in the question.))

I thought that was a great question, as much for where it led as for the puzzle it posed. What did you make of it?