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14 problems match “inequalities”. Clear

Maximum integer k in a derivative inequality ●●●●○ 100 ptsLet $f(x)=e^{x}-x-2$ . For $x>0$ , the inequality $(x-k)f'(x)+x+1>0$ always holds, where $k$ is an integer. Find the maximum value of $k$ .
Minimum ratio of derivative energy to squared mean ●●●●○ 100 ptsLet $f$ be a continuously differentiable real-valued function on $[0,1]$ with $\int_{1/3}^{2/3} f(x)\,dx = 0$ . Find the minimum value of $\dfrac{\int_0^1 (f'(x))^2\,dx}{\left(\int_0^1 f(x)\,dx\right)^2}$ over all such $f$ for which the denominator is nonzero.
Longest chain of quadratic equations ●●●●○ 100 ptsQuadratic equations are solved one after another. Each equation has leading coefficient 1, of the form $x^2+p_jx+q_j=0$ . Whenever an equation has two distinct real roots, the next equation is formed as follows: its constant term is the larger root of the current equation, and its linear coefficient is the smaller root of the current equation. The process stops at the first equation that does not have two distinct real roots. Find the maximum possible number of quadratic equations in such a sequence.
Maximum of |bc| for a bounded quadratic ●●●●● 200 ptsLet complex numbers $a$ , $b$ , $c$ satisfy: for any complex number $z$ with $|z|\leq 1$ , we have $|az^2+bz+c|\leq 1$ . Find the maximum value of $|bc|$ . Round to two decimal places.