Let a_2 , a_3 R such that | a_2 - a_3 | = 6 and f ( x ) = | * 20 … - Vidyadip

MCQ

Let ${a_2},{a_3} \in R$ such that $\left| {{a_2} - {a_3}} \right| = 6$ and $f\left( x \right) = \left| {\begin{array}{*{20}{c}}
1&{{a_3}}&{{a_2}}\\
1&{{a_3}}&{2{a_2} - x}\\
1&{2{a_3} - x}&{{a_2}}
\end{array}} \right|,x \in R.$ Then the greatest value of $f(x)$ is

A
$36$
B
$24$
C
$12$
✓
$9$

✓

Answer

Correct option: D.

$9$

d
Apply $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ and $\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}$

$f(x)=-x^{2}+\left(a_{2}+a_{3}\right) x-a_{2} a_{3}$

$\left|a_{2}-a_{3}\right|=\frac{\sqrt{D}}{|a|}=6=\sqrt{D}=6$

$\max .$ value $=-\frac{D}{4 a}=9$

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