I'm trying to determine, whether the function:
$\frac{2\sin x-5\cos x}{2x^{2}+3}$ is bounded.
These are the steps I followed: $\left |\frac{2\sin x}{2x^{2}+3} \right | + \left | \frac{5\cos x}{2x^{2}+3} \right | \leq \left | \frac{2}{2x^{2}+3} \right|*1+\left | \frac{5}{2x^{2}+3} \right |*1$
And I put $0$ for $x$ to get the upper bound. Then I got $\leq \frac{2}{3}+\frac{5}{3}$. However, that upper bound is incorrect.
Can you tell me where I did a mistake? Also how would I find the lower bound?
Well, the mistake is when you put it to zero as it isn't the maximum of the equation.
Rewrite the equation using $a\sin(x)+b\cos(x) = \sqrt{a^2+b^2}\sin(2x)$ and you can derive and find the maximum of the function (think it is how it should be done).