If f(x) = $\frac{x}{4x^2+7x+9}$ , for all x > 0, then what is the greatest value of f(x)?

<p>f (x) =&nbsp;<math><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><mn>4</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mfrac></mrow></math>&nbsp;………. (1)</p><p>Divide by x in numerator and denominator equation (1) can be written as f (x) =&nbsp;<math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></mfrac></mrow></math></p><p>f (x) is maximum, when the denominator is minimum we have to find the minimum value of&nbsp;<math><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></math>&nbsp;, so, denomination will also minimum</p><p>we know that, AM ≥ GM</p><p><math><mrow><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></mfrac><mo>≥</mo><mroot><mrow><mn>4</mn><mi>x</mi><mo>×</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><mrow></mrow></mroot></mrow></math></p><p><math><mrow><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></mfrac><mo>=</mo><mn>6</mn></mrow></math></p><p><math><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mn>2</mn></mrow></math></p><p>The minimum value of&nbsp;<math><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></math>&nbsp;is 12</p><p>Therefore, the maximum value of f (x) =&nbsp;<math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>2</mn><mo>+</mo><mn>7</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>9</mn></mrow></mfrac></mrow></math></p>

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