Matematické Fórum

stuart clark

$(1)$ If $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(x)$ is differentiable function and $|f(x)|\leq \cos^2(x^2).$ Then minimum number of roots of $f'(x)=0$ in $x\in (0,2\sqrt{\pi})$ is

$(2)$ If $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(x)$ is differentiable function and $|f(x)|\{x\}\leq \cos^2(x).$ Then minimum number of roots of $f'(x)=0$ in $x\in (0,3\pi$ is

$(3)$ If $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(x)$ is differentiable function and $|f(x)|\{x\}\leq \cos^2(x).$ Then minimum number of roots of $f(x)=0$ in $x\in (0,3\pi)$ is

where $\{x\}=x-\lfloor x \rfloor$ (fractional part of $x$ )

jardofpr · 26. 09. 2019 12:00

hi ↑ stuart clark:

at first sight it looks to me that

(1)

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(2)

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(3)

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stuart clark · 28. 09. 2019 19:24

Thanks ↑ jardofpr:

stuart clark

Please have a look on

Find the largest constant $k$ for which $\sum^{4}_{i=1}\bigg(x_{i}+\frac{1}{x_{i}}\bigg)^{3}\geq k$ for all $x_{1},x_{2},x_{3},x_{4}>0$ such that $x^{3}_{1}+x^{3}_{3}+3x_{1}x_{3}=x_{2}+x_{4}=1$

jardofpr · 29. 09. 2019 12:43

hi ↑ stuart clark:

before the hints to solution, I'd like just to note that it is good practice (and the forum rule) to insert
new problem as new thread; the exceptions are the dynamic threads designed specially for the purpose
of rotating practice problems (like the one here limit marathon )

to the problem you stated

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