In the attached image I have a problem that asks to find a function greater than the function given. Now I am not sure if I am correct, but is Sin(x) or Cos(x) a solution to the problem?
2026-04-07 03:20:25.1775532025
Are Sin(x) and Cos(x) functions that are greater than the function (Cos^2(x)) / (1+x^2)?
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Recall: $-1 \le \cos(x) \le 1$ for all $x$ -- this is especially obvious if you graph the function. If you were to square it, then, clearly, $0 \le \cos^2(x) \le 1$. This why the hint referenced the "range of trig. functions" -- think of the values the functions map to, and try to get a function which is not only simpler, but greater in value.
Thus, clearly, a good choice for $g$ would be $1/(1+x^2)$, as
$$\frac{\cos^2(x)}{1+x^2} \le \frac{1}{1+x^2} = g(x)$$
Moreover, this choice of $g$ works nicely since $g(x)$ has a familiar antiderivative, making it worthwhile for our intended purposes.