The minimum value of the expression 9x2sin 2x-4-xsin x for x - 0- is

The correct option is D 12 We know that, AP≥ GP ⇒3x sin x +22≥√((3xsin x)2) Squaring both sides 9x^2sin^2x+4+12xsin x4≥ 6xsin x⇒ 9x^2sin^2x+ ...

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Standard XII

Mathematics

Property 2

The minimum v...

Question

The minimum value of the expression $\frac{9 x^{2} {sin}^{2} x + 4}{x sin x}$ for $x ε (0, π)$ is

\frac{16}{3}

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12

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6

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\frac{8}{3}

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Solution

The correct option is D $12$
We know that,
$A P \geq G P$ $\Rightarrow \frac{3 x sin x + 2}{2} \geq \sqrt{(3 x sin x) 2}$
Squaring both sides
$\frac{9 x^{2} {sin}^{2} x + 4 + 12 x sin x}{4} \geq 6 x sin x \Rightarrow 9 x^{2} {sin}^{2} x + 4 + 12 x sin x \geq 24 x sin x \Rightarrow 9 x^{2} {sin}^{2} x + 4 \geq 12 x sin x \Rightarrow \frac{9 x^{2} {sin}^{2} x + 4}{x sin x} \geq 12$

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The minimum value of the expression 9x2sin2x+4xsinx for xε(0,π) is

The correct option is D 12We know that,AP≥GP ⇒3xsinx+22≥√(3xsinx)2Squaring both sides9x2sin2x+4+12xsinx4≥6xsinx⇒9x2sin2x+4+12xsinx≥24xsinx⇒9x2sin2x+4≥12xsinx⇒9x2sin2x+4xsinx≥12

The minimum value of the expression $\frac{9 x^{2} {sin}^{2} x + 4}{x sin x}$ for $x ε (0, π)$ is