For all real values of $x$ , the maximum value of the expression $\frac{x}{x^{2} - 5 x + 9}$ is

1

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45

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90

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None of these

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Solution

The correct option is A $1$
$y = \frac{x}{x^{2} - 5 x + 9}$
$\Rightarrow y x^{2} - (5 y + 1) x + 9 y = 0$
As $x$ is real, discriminant $\geq 0$
$\Rightarrow (5 y + 1)^{2} - 36 y^{2} \geq 0$
$\Rightarrow 11 y^{2} - 10 y - 1 \leq 0$
$∴ y \in [- \frac{1}{11}, 1]$
i.e. Maximum value of expression is $1$ .
Hence, option A.

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