Inegalitate integrala de tip Cauchy si cu derivata
Posted: Fri Mar 21, 2008 12:44 am
Fie \( f:[0,1]\to\mathbb{R} \) o functie de clasa \( C^{1} \) pe intervalul \( [0,1] \). Sa se demonstreze ca
\( \int_0^1f^{2}(x)dx-\left(\int_0^1f(x)dx\right)^{2}\leq \int_0^1|f^{\prime}(x)|dx\left(M-\int_0^1f(x)dx\right) \),
unde \( M=\sup_{x\in [0,1]}f(x) \).
\( \int_0^1f^{2}(x)dx-\left(\int_0^1f(x)dx\right)^{2}\leq \int_0^1|f^{\prime}(x)|dx\left(M-\int_0^1f(x)dx\right) \),
unde \( M=\sup_{x\in [0,1]}f(x) \).
