Întrebare Valoare minima

Buna ziua
Considerand ca ecuatia P(x)=0,unde:
\[P(x)=x^n-a_1x^{n-1}+....+a_{n-1}x+a_n,are\ radacinile\ x_i,\\ cu\ i\in\overline{1,n},reale\ si\ distincte,\ sa\ se\ arate\ ca\ pentru\\ (\forall)x\in R,x>max\{x_1,x_2,x_3.....x_n\}si\ pentru\\ \forall n\in N^{*},valoarea\ minima\ a\ expresiei\\ \dfrac{(nx-a_1)P'(x)}{P(x)}\ este:\\ a)n^2;b)2;c)a_1;d)n;e)\sqrt(n)\]
ma gandesc ca la extrem prima derivata se anuleaza dar mai departe?
multumesc

Buna ziua,

Pe scurt: \(\frac{P'\left ( x \right )}{P\left ( x \right )}=\frac{1}{x-x_{1}}-\frac{1}{x-x_{2}}+...+\frac{1}{x-x_{n}}\).

Conform relatiilor lui Viete: \(x_{1}+x_{2}+...+x_{n}=a_{1}\) \(\Rightarrow\)

\( nx-a_{1}=\left ( x-x_{1} \right )+\left ( x-x_{2} \right )+...+\left ( x-x_{n} \right )\).

Cum \(x>max\left \{ x_{1},x_{2},...,x_{n} \right \}\Rightarrow x-x_{i}>0,\: \forall i\in \left \{ 1,2,...,n \right \}\).

Atunci \(\frac{\left ( nx-a_{1}\right )\cdot P'\left ( x \right )}{P\left ( x \right )}\) ia forma urmatoare:

\(\left [ \left ( x-x_{1} \right )+\left ( x-x_{2} \right )+...+\left ( x-x_{n} \right ) \right ]\cdot \left ( \frac{1}{x-x_{1}}+\frac{1}{x-x_{2}}+...+\frac{1}{x-x_{n}} \right )\)

Folosind inegalitatea :

\(\left ( u_{1}+u_{2}+...+u_{n} \right )\cdot \left ( \frac{1}{u_{1}}+\frac{1}{u_{2}}+...+\frac{1}{u_{n}} \right )\geq n^{2}\), \(u_{i}>0,\: i\in \left \{ 1,2,...,n \right \}\)

avem: \(\frac{\left ( nx-a_{1}\right )\cdot P'\left ( x \right )}{P\left ( x \right )}\) \(\geq n^{2}\).

Raspuns corect: \(a)\).

Buna ziua
Va multumesc pentru atentie.
Daca nu va suparati o intrebare:nu stiu de unde vine expresia:
\[\dfrac{P'(x)}{P(x}=\dfrac{1}{x-x_1}+\dfrac{1}{x-x_2}+\dots+\dfrac{1}{x-x_n}\]
multumesc

Derivati functia polinomiala \(P\) sub forma factorizata:

\(P\left ( x \right )=\left ( x-x_{1} \right )\cdot \left ( x-x_{2} \right )\cdot ...\cdot \left ( x-x_{n} \right )\) - ce obtineti?

(Cum ati deriva un produs de \(3\) functii, spre exemplu \(\left (f\cdot g\cdot h \right )'=?\)
Cat este atunci \(\left [\left ( x-x_{1} \right )\cdot \left ( x-x_{2} \right )\cdot \left ( x-x_{3} \right ) \right ]' ?\)
Ce observati? Cum derivati un produs de 4 functii? Si, in final, cum derivati \(P\) ?)

Daca sunteti gata, puteti calcula si \(\frac{P'\left ( x \right )}{P\left ( x \right )}\).

Buna ziua
acum am inteles totul.
Dar ma gandesc asa: este intr-un fel greu sa reproduc,dar mi-te sa le creiezi cum faceti Dvs?
multumesc