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Funcţiile \( f \), \( f^{\prime} \), \( f^{\prime\prime} \) sunt continue pe \( [0,a] \), \( a \geq 0 \) şi \( f(0)=f^{\prime}(0)=0 \). Să se arate că \( \int_0^a|f(x)f^{\prime\prime}(x)|dx \leq \frac{a^2}{2}\int_0^a{(f^{\prime\prime}(x))^2}dx \).

Concursul Naţional "Traian Lalescu", 2008, profil electric, anul I

Observaţie: În enunţul original, inegalitatea cerută era \( \int_0^a|f(x)f^{\prime}(x)|dx \leq \frac{a^2}{2}\int_0^a{(f^{\prime\prime}(x))^2}dx \).

Here a version for functions of class \( C^{n+1} \):

Let \( a>0 \) and f a function from [0;a] in R, \( f\in C^{n+1} \),
with \( f(0)=f^{\prime} (0)=...=f^{(n)}(0)=0 \).

Prove that

\( \int_{0}^{a}|ff^{(n+1)}|\leq \frac{a^{n+1}}{n!\sqrt{(2n+1)(2n+2)}}\int_{0}^{a}(f^{(n+1)})^2 \).