question 17 (1 point) which of the following profit functions have break - even points at x = 2 and x = 9…

question 17 (1 point) which of the following profit functions have break - even points at x = 2 and x = 9? a) $p(x)=-3(x + 2)(x + 9)$ b) $p(x)=-3(x + 2)(x - 9)$ c) $p(x)=-3(x^{2}-11x + 18)$ d) $p(x)=-3x^{2}-33x - 54$ page 17 of 35

question 17 (1 point) which of the following profit functions have break - even points at x = 2 and x = 9? a) $p(x)=-3(x + 2)(x + 9)$ b) $p(x)=-3(x + 2)(x - 9)$ c) $p(x)=-3(x^{2}-11x + 18)$ d) $p(x)=-3x^{2}-33x - 54$ page 17 of 35

Answer

Explanation:

Step1: Recall break - even point concept

Break - even points occur when $P(x)=0$. If the break - even points are at $x = 2$ and $x=9$, then $(x - 2)$ and $(x - 9)$ are factors of the profit function $P(x)$.

Step2: Analyze each option

  • Option a: $P(x)=-3(x + 2)(x + 9)$ has break - even points at $x=-2$ and $x=-9$.
  • Option b: $P(x)=-3(x + 2)(x - 9)$ has break - even points at $x=-2$ and $x = 9$.
  • Option c: Expand $P(x)=-3(x^{2}-11x + 18)=-3(x - 2)(x - 9)$. When $P(x)=0$, $-3(x - 2)(x - 9)=0$. Using the zero - product property, $x=2$ or $x = 9$.
  • Option d: First, factor out $-3$ from $P(x)=-3x^{2}-33x - 54=-3(x^{2}+11x + 18)=-3(x + 2)(x+9)$ which has break - even points at $x=-2$ and $x=-9$.

Answer:

C. $P(x)=-3(x^{2}-11x + 18)$