Function analysis #1 – Min of function via expression

Brief explanation: Here we analyse the minimum of a multivariable function without the ‘normal’ methods: just arranging the expression in a different way.

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Question: Find the minimum of the function:

enunciado.png— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Comprehending: The nice thing about this problem is that you don’t even need to know multivariable calculus, you just need to look critically to the expression.

First of all, what do we expect from such an expression? Remember that a function of type ax² + bx + c is a parabola, so what do we expect from a similar function in 3d? The expression indicates parabolas in both x and y axis, so we get the following graph:

capa.png

I think the form of the graph is kind of intuitive, if you look at it like I did above. As you can see, there is a minimum value of the function: that is what we want to find out.

When you do this kind of exercises, try to visualize the graph in your mind, without actually plotting it on you pc or calculator. I only inserted it here in case you did not know how this would turn out.

So, how do we calculate the minimum? The technique we’ll be using is called complete the square. This just means transforming the expression into the product of the sum of two numbers, like the following example:

caso notavel.png

This is the most basic case. A more difficult one would be:

caso notavel 2.png

Now let’s try with the one in the question.

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Solving: 

The one in the question is actually already neatly organized for us, meaning we have an x part and a y part.

1.png

Now we think: the x^2 part is obtained by the multiplication of the x’s when we square the expression. What about the 3x part? If you look at the first example I gave about completing the square, you’ll see that it is the equivalent to the 2xy part, meaning it will be the result of the multiplication of x with a number we’ll find out. The y part is easier, because the numbers already suggest the square, probably multiplied externally by some number.

It is hard to explain this kind of ‘backwards thinking’. If you have doubts, analyze the following expression until you are sure you understand how this was done:

2.png

Again, reverse engineer the expression if you have doubts. This is the ‘trick’ we need to solve the problem.

Think about the expression we got. What do we know about it? Well, everything that is squared is always positive, right? This must mean that the minimum value is obtained when these parts are zero (their minimum). The outside part will be the value of the function at this point.

All we need is to equal the squared parts to zero! This is easy:

43

These are the coordinates of the minimum of the function! When we plug the values into the equation:

5.png

Which is the value of the function at the minimum!

Quite easy, right? When you play smart, there’s no need for more complicated methods!

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