I'm having trouble with the following question:

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> Show $|ax + by| \leq (a^{2} + b^{2})^{1/2}$ if $x^{2} + y^{2} = 1$ by finding the maximum and minimum values of $f(x, y) = ax + by$ on the unit circle. 
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> 
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I'm learning multivariable calculus on my own, and I came across this question. I've been struggling for a couple of hours, and I'd really appreciate some help. I approach this problem using Lagrange Multipliers.

Here's what I've tried so far:

Let $f(x, y) = ax + by$ and let $g(x, y) = x^2 + y^2 - 1$. Then, by lagrange multiplier method, we have

$$a = \lambda(2x) $$

and 

$$b = \lambda(2y),$$

from which we get

$$\lambda = \frac{a}{2x} = \frac{b}{2y}.$$

Then, I divided the two equations and I found $x = \frac{ay}{b}$, which I plugged into the constraint equation, but I got nowhere. Can someone please help me with this problem?