Question

If $f(x)= \begin{cases}x^{2}-1, & 0 < x < 2 \\ 2 x+3, & 2 \leq x < 3\end{cases}$
the quadratic equation whose roots are $\displaystyle\lim _{x \rightarrow 2^{-}} f (x)$ and $\displaystyle\lim _{x \rightarrow 2^{+}} f (x)$ is

• A. $x^{2}-14 x+49=0$
• B. $x^{2}-6 x+9=0$
• C. $x^{2}-10 x+21=0$
• D. $x^{2}-7 x+8=0$

Answer 1

Answer: (C)

$\alpha=\displaystyle\lim _{x \rightarrow 2^{-}} f(x)=\displaystyle\lim _{x \rightarrow 2} x^{2}-1=3$
$\beta=\displaystyle\lim _{x \rightarrow 2^{+}} f(x)=\displaystyle\lim _{x \rightarrow 2} 2 x+3=7$
$x^{2}-(\alpha+\beta) x+\alpha \beta=0$
$x^{2}-10 x+21=0$