Animated Solution for Mathematics - Trigonometry: The number of values of $x$ where the function $f(x)=\cos x+\cos (\sqrt{2}x)$ attains its maximum is

Imagine you are standing on a bridge, watching two waves ripple across a pond. One wave is slow and steady, while the other, driven by a strange, irrational frequency, seems to dance to a different rhythm.

This is exactly the physical intuition behind the function $f(x)=\cos x+\cos (\sqrt{2}x)$. We want to know how many times these two waves can perfectly align at their highest peak. Let's dive into the mathematics of this synchronization.

First, let's establish our goal. We know that for any angle $\theta$, the maximum value of $\cos \theta$ is $1$.

Since our function $f(x)$ is the sum of two such terms, the absolute maximum value it can possibly attain is $1+1=2$. This is our 'ceiling'.

For the function to reach this ceiling, both $\cos x$ and $\cos (\sqrt{2}x)$ must simultaneously equal $1$. If even one of them falls short, the sum will be less than $2$. This is the crucial constraint: we are looking for the intersection of two sets of solutions.

Let's break down the conditions for each wave to reach its peak. For $\cos x=1$, we know the solutions are $x=2n\pi$, where $n$ is any integer. These are the points where the first wave hits its maximum.

Now, consider the second wave: $\cos (\sqrt{2}x)=1$. This occurs when the argument $\sqrt{2}x$ is an even multiple of $\pi$, say $2m\pi$, where $m$ is another integer.

Solving for $x$, we get:

Now, for the function to reach its maximum of $2$, these two conditions must be met at the same value of $x$. So, we set them equal:

This is where the beauty of the problem reveals itself. If we divide both sides by $\pi$, we get $2n=\sqrt{2}m$.

Rearranging this gives us the ratio:

Stop and think about this for a moment. On the left side, we have the ratio of two integers, $m$ and $n$, which must be a rational number. On the right side, we have $\sqrt{2}$, which is famously irrational.

A rational number can never equal an irrational number! This creates a logical wall. The only way to bypass this wall is if the equation holds true in a way that doesn't involve a non-zero ratio. That only happens when $m=0$ and $n=0$.

Substituting $n=0$ back into our expression for $x$, we find $x=2(0)\pi =0$. At $x=0$, both $\cos (0)$ and $\cos (0)$ are $1$, and their sum is $2$.

For any other integer $n$ or $m$, the waves will never align at their peak again. The irrationality of $\sqrt{2}$ ensures that the two waves are perpetually out of phase.

Thus, there is exactly one value of $x$—namely $x=0$—where the function attains its maximum. It is a singular, elegant point of perfect synchronization in an otherwise chaotic, out-of-sync universe.