Edercise:
On Considere la fonction \( f \) definie par:
\[
f(x)=\frac{2 x^{2}-2 x+1}{x^{2}-x}
\]
1) Determiner Dof puus les reils \( a, b \). et \( C \) telque:
و \( f(x)=a+\frac{b}{x}+\frac{c}{x-1}: i[1] \)
21a) Donner le tablean de Variation def « pour Calculer la derivé de \( f(x) \) on pent Utiliser l'ecritire \( [1] \gg \) b) en deduire que \( \forall x \in] 0,1 \) I

Answer

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Answer

\( f(x) = 2 + \frac{x^2 - x + 1}{x(x - 1)} = a + \frac{b}{x} + \frac{c}{x - 1} \)

Steps

Step 1 ：\( f(x) = \frac{2x^2 - 2x + 1}{x^2 - x} = \frac{2(x^2 - x) + (x^2 - x + 1)}{x^2 - x} \)

Step 2 ：\( f(x) = \frac{2(x^2 - x)}{x^2 - x} + \frac{x^2 - x + 1}{x^2 - x} = 2 + \frac{x^2 - x + 1}{x^2 - x} \)

Step 3 ：\( f(x) = 2 + \frac{x^2 - x + 1}{x(x - 1)} = a + \frac{b}{x} + \frac{c}{x - 1} \)