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 35 Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$. 27 Does there exist a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$ such that $\mathbf{A}^{2}=-\mathbf{I}$? 23 How to show that $f'(x)<2f(x)$ 20 If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$ 14 On the inequality $\int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$

### Reputation (12,867)

 +10 On the inequality $\int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$ +10 If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$ +10 How find this$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+…+\frac{1}{{{p}_{n}}}<10$ +10 Riemann surface arising as a quotient of the upper half-plane.

### Questions (3)

 25 Is the image of a nowhere dense closed subset of $[0,1]$ under a differentiable map still nowhere dense? 23 Is the image of a Borel subset of $[0,1]$ under a differentiable map still a Borel set? 8 Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$?

### Tags (107)

 196 real-analysis × 63 80 matrices × 12 137 complex-analysis × 42 78 integration × 21 131 calculus × 28 78 polynomials × 9 91 linear-algebra × 24 74 inequality × 21 84 analysis × 24 55 measure-theory × 18

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