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Exercise: Define $$f(x) = x^3 - \sin^2(x)\tan(x)$$ $$g(x) = 2x^2 - \sin^2(x) -x\tan(x)$$ Find out, for each of these two functions, whether it is positive or negative for all $x \in (0, \pi/2)$, or whether it changes sign. Prove your answer.

The problem

The standard solution to this exercise requires to compute the derivatives of $f$ and $g$ as many times as six and five respectively, however I think that such an answer is neither illuminating nor typical for Rudin's exercises. While trying to solve it by myself I would have guessed the solution to rely on some clever inequality involving sine and tangent. In particular, I have noticed that the presence of $x^3$ and $2x^2$ is not random and this is something that the iterated derivatives do not highlight. Indeed, by the Mean Value Theorem, one can link the behavior of sine and tangent of $x$ to that of $x$ multiplied by some constant, so that, for instance, $\sin^2x\tan x$ behaves like a multiple of $x^3$.

I would like to know whether there is an actual elegant solution or if this exercise were mere computation.


As always, any comment or answer is much appreciated and let me know if I can explain myself clearer!

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  • $\begingroup$ Think about Taylor. The result is immediate $\endgroup$ Sep 8, 2022 at 11:37
  • $\begingroup$ @ClaudeLeibovici I have actually tried using Taylor (with Lagrange’s Remainder) but to no avail. Could you specify what did you have in mind? $\endgroup$ Sep 8, 2022 at 11:49
  • $\begingroup$ The term $\tan(x)$ takes the functions to $-\infty$. How does they behave close to $0$ ? $\endgroup$ Sep 8, 2022 at 12:05
  • $\begingroup$ @ClaudeLeibovici Close to 0 both functions converge to 0, however the functions may still change sign between 0 and $\pi/2$. $\endgroup$ Sep 8, 2022 at 12:17
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    $\begingroup$ For sure but the basic series are valid for all $x$; there is an explicit formula for the coefficients and ou can prove that, by thier definition, they are all negative. QED $\endgroup$ Sep 8, 2022 at 12:34

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