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Math  /  Algebra

Question17 Soit f(x)=(1−m)x2+2mx−m+1f(x)=(1-m) x^{2}+2 m x-m+1 où mm est un paramètre réel différent de 1,x11, x_{1} et x2x_{2} les racines quand elles existent. 1∘1^{\circ} Calculer mm pour que x1x_{1} et x2x_{2} existent . 2∘2^{\circ} Calculer mm pour que x1x_{1} et x2x_{2} quand elles existent, aient le même signe . 3∘3^{\circ} a) Montrer que 1x1−1+1x2−1\frac{1}{x_{1}-1}+\frac{1}{x_{2}-1} est un nombre constant différent de mm. b) Calculer mm pour que x1(x2+1)+x1<mx_{1}\left(x_{2}+1\right)+x_{1}<m. 4∘4^{\circ} Calculer mm pour que f(x)>0f(x)>0 pour tout xx.

Studdy Solution
Determine conditions for f(x)>0 f(x) > 0 for all x x . This requires:
- The quadratic has no real roots (Δ<0\Delta < 0). - The leading coefficient a=1−m>0 a = 1-m > 0 .
The solution for each part of the problem will depend on solving the inequalities and evaluating expressions as described in each step.

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