Question

The graph of $f(x)=a{x}^2+bx+c$ is given, for which $\ell (AB)=4,\ell (AC)=4$ and $b^2-4ac=-8.$

Range of $h(x)=\left(\frac{3}{2}+a\right)x^2+(b-1)x+(c-6)=0$ when $x\in [-2,0]$ is:

• A. $\left[\frac{39}{8},6\right]$
• B. $\left[\frac{39}{8},8\right]$
• C. $[6,8]$
• D. $\left[\frac{39}{8},8\right)$

Answer 1

Answer: (B)

$\frac{-D}{4a}=4\Rightarrow \frac{8}{4a}=4\Rightarrow a=\frac{1}{2}$
$\frac{-b}{2a}=-4\Rightarrow b=8a\Rightarrow b=4$
$b^2-4ac=-8\Rightarrow b=8a\Rightarrow b=4$
$b^2-4ac=-8\Rightarrow 16-2c=-8$
$\Rightarrow c=12$
$f(x)=\frac{x^2}{2}+4x+12$
$h(x)=2x^2+3x+6$

Range of $h(x)$ is $\left[\frac{39}{8},8\right]$