Do NOT Solve This With Calculus!

At first glance, this maths puzzle appears to require multivariable calculus.

But if we go down that route, this becomes an uninteresting textbook exercise.

Instead, I want to challenge you to think algebraically and geometrically.

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So if you’d like to solve more challenging maths questions, please go take a look. Thank you.

Solution

Notice in each of the two radicals, there are x, x², y, y² and constant terms.

So we can complete the square for all of them, let’s do that.

first radical

second radical

This means our expression now becomes

Let’s pause here for a second and look at the following diagram.

Notice the two arbitrary points P1 and P2, as well as the points (2, 3) and (4, 4).

Here we see that the expression we want to minimise is the sum of shortest distances an arbitrary point P makes with (2, 3) and (4, 4).

Well at this point, it should be obvious to see that we minimise the distance if the point P is anywhere along the straight line connecting (2, 3) and (4, 4)!

It doesn’t matter where P is as long as it lies on the line because the sum of the two distances is always the shortest distance between the points (2, 3) and (4, 4).

And we can find that using the distance formula, giving us the minimum value of our expression.

And that’s our answer. How amazing :)

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