Question

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

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)

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

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