Found problems: 8
Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that
$$ \left| \int_0^2 f(t) dt\right| >1. $$
Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$
Show that
[b]a)[/b] $ f(n)=0. $
[b]b)[/b] $ f $ is globally nonnegative.
[i]Gheorghe Grigore[/i]
Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a countinuous function such that
$$ \int_0^1 f(x)dx=\int_0^1 xf(x)dx. $$
Show that there is a $ c\in (0,1) $ such that $ f(c)=\int_0^c f(x)dx. $
[b]a)[/b] Let $ h:\mathbb{R}\longrightarrow\mathbb{R} $ he a function that admits a primitive $ H $ such that the function $ h/H $ is constant. Prove that there is a real number $ \gamma $ such that $ h(x)=\gamma\cdot\exp \left( x\cdot\frac{h}{H} (x) \right) , $ for any real number $ x. $
[b]b)[/b] Find the functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ that admit the primitives $ F,G, $ respectively, that satisfy $ f=\frac{G+g}{2},g=\frac{F+f}{2} $ and $ f(0)=g(0)=0. $
Let $ F $ be the primitive of a continuous function $ f:\mathbb{R}\longrightarrow (0,\infty ), $ with $ F(0)=0. $
Determine for which values of $ \lambda \in (0,1) $ the function $ \left( F^{-1}\circ \lambda F \right)/\text{id.} $ has limit at $ 0, $ and calculate it.
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admit a primitive $ F $ defined as $ F(x)=\left\{\begin{matrix} f(x)/x, & x\neq 0 \\ 2007, & x=0 \end{matrix}\right. . $
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that:
$$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$
Prove the rationality of the number $ \frac{1}{\pi }\int_{\sin\frac{\pi }{13}}^{\cos\frac{\pi }{13}} \sqrt{1-x^2} dx. $