Multiple choice

Let $\alpha$ & $\beta$ be the roots of the equation, $ax^{2}+bx+c=0$ where $1<\alpha<\beta$, then $\displaystyle\lim_{x\to m}\frac{|ax^{2}+bx+c|}{ax^{2}+bx+c}=1$, then which of the following is correct?

  1. $\displaystyle a>0$ & $m>1$
  2. $\displaystyle a<0$ & $m<1$
  3. $\displaystyle a<0$ & $\alpha < m < \beta$
  4. $\displaystyle \frac{|a|}{a}=1$ & $m>\alpha$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The limit of |f(x)|/f(x) is 1 if f(x) > 0. For the quadratic ax^2+bx+c to be positive between its roots alpha and beta, the parabola must open downwards (a < 0). Thus, if alpha < m < beta and a < 0, the expression is positive.

AI explanation

For the absolute value expression $|ax^2 + bx + c| / (ax^2 + bx + c)$ to equal 1 as $x$ approaches $m$, the quadratic expression $ax^2 + bx + c$ must be positive at that value. Given the roots $1 < \alpha < \beta$, the parabola is positive outside the interval $(\alpha, \beta)$ if $a > 0$, or inside the interval $(\alpha, \beta)$ if $a < 0$. Because the question specifies that $m$ can be evaluated in the region where the function's sign matches its absolute value, the accurate condition is that $m$ resides outside the roots if $a > 0$ or between the roots if $a < 0$. Thus, stating $a < 0$ and $\alpha < m < \beta$ is merely one valid scenario, making a generalized statement about $m$'s relation to the roots based on the sign of $a$ the true mathematical condition.

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