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Found problems: 2

1974 Putnam, B2
Tags: continuous and differentiable
Let $y(x)$ be a continuously differentiable real-valued function of a real variable $x$. Show that if $y'(x)^2 +y(x)^3 \to 0$ as $x\to \infty,$ then $y(x)$ and $y'(x) \to 0$ as $x \to \infty.$

1980 Putnam, A6
Tags: continuous and differentiable , function
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1 .$ Determine the largest real number $u$ such that $$u \leq \int_{0}^{1} |f'(x) -f(x) | \, dx $$ for all $f$ in $C.$