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I have a function that I would like help finding the maximum value for, without a graph. The function is:

$$ y = 5\sin(3000t) + 10\sin(2000*\pi*t) $$

When I tried Newtons method and equated it to $15$, there was no solution. When I found the derivative and equated to $0$,it did not work as there are many local maxima and minima, so when newtons method is applied, the answer depends on the starting value.

Any help on how to tackle this problem?

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  • $\begingroup$ $t$ is defined in the whole $\mathbb{R}$ ?? $\endgroup$ – Nizar Nov 3 '15 at 7:59
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    $\begingroup$ $\frac{dy}{dt}$ $\endgroup$ – hjpotter92 Nov 3 '15 at 8:01
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Since $3000$ and $2000\pi$ are incommensurable, i.e., not rational multiples of each other, the values of this function will be dense in the interval $[-15,15]$ but will have no maximum.

At the maxima of the second term, when $2000\pi t=\frac\pi2+k2\pi$, the function has the values $$ 5\sin(\tfrac34\pi+3k)+10 $$ and it is well known that $\sin(a+bk)$, $k\in\Bbb Z$ is dense in $[-1,1]$ if and only if $b$ is not a rational multiple of $\pi$.

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Consider $t$ of the form $t=n+\frac{1}{4000}$, where $n$ ranges over the natural numbers. Then the second term is $10$.

The first term is $5\sin(3000n+\frac{3}{4})$. Since $\pi$ is irrational, the numbers $3000n$ modulo $2\pi$ are dense in the interval $[0,2\pi)$. It follows that the supremum of the $5\sin(3000n+\frac{3}{4})$ is $5$.

Our function therefore has supremum $15$. It does not have a maximum.

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