Question

Let $a_2, a_3 \in R$ be such that $\left|a_2-a_3\right|=6$.
Let $f(x)=\begin{vmatrix} 1 & a_3 & a_2 \\ 1 & a_3 & 2 a_2-x \\ 1 & 2 a_3-x & a_2 \end{vmatrix}, x \in R$
The maximum value of $f(x)$ is

• A. 6
• B. 9
• C. 12
• D. 36

Answer 1

Answer: (B)

Using $R_2 \rightarrow R_2-R_1, R_3 \rightarrow R_3-R_1$, we get
$f(x) =\begin{vmatrix}1 & a_3 & a_2 \\0 & 0 & a_2-x \\0 & a_3-x & 0\end{vmatrix} $
$=-\left(a_2-x\right)\left(a_3-x\right)=-\left[x^2-\left(a_2+a_3\right) x+a_2 a_3\right] $
$ =\frac{1}{4}\left(a_2-a_3\right)^2-\left(x-\frac{a_2+a_3}{2}\right)^2 \leq 9$
$f(x)$ attains maximum value 9 when
$x=\frac{1}{2}\left(a_2+a_3\right)$