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Maths-

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Question

The maximum and minimum values of $y equals fraction numerator a x to the power of 2 end exponent plus 2 b x plus c over denominator A x to the power of 2 end exponent plus 2 B x plus C end fraction$ are those for which

$a x^{2} + 2 b x + c - y (A x^{2} + 2 B x + C)$ is equal to zero
$a x^{2} + 2 b x + c - y (A x^{2} + 2 B x + C)$ is a perfect square
$\frac{d y}{d x} = 0$ and $\frac{d^{2} y}{d x^{2}} = 0$
$a x^{2} + 2 b x + c - y (A x^{2} + 2 B x + C)$ is not a perfect square

The correct answer is: $a x to the power of 2 end exponent plus 2 blank b x plus c minus y open parentheses A x to the power of 2 end exponent plus 2 blank B x plus C close parentheses$ is a perfect square

Let $f open parentheses x close parentheses equals fraction numerator a x to the power of 2 end exponent plus 2 b x plus c over denominator A x to the power of 2 end exponent plus 2 B x plus C end fraction$
Let $P left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ be a point on the curve $y equals f left parenthesis x right parenthesis$ where $y$ is maximum or minimum. Then, at $P$

$fraction numerator d y over denominator d x end fraction equals 0$ and $fraction numerator d to the power of 2 end exponent y over denominator d x to the power of 2 end exponent end fraction not equal to 0$

The equation of the tangent at $P left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ is

$y minus y subscript 1 end subscript equals open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript P end subscript open parentheses x minus x subscript 1 end subscript close parentheses rightwards double arrow y equals y subscript 1 end subscript open square brackets because open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript P end subscript equals 0 close square brackets$

Putting $y equals y subscript 1 end subscript$ in $y equals fraction numerator a x to the power of 2 end exponent plus 2 b x plus c over denominator A x to the power of 2 end exponent plus 2 B x plus C end fraction$ , we get the $x$ -coordinate of $P$ . This means that $y equals fraction numerator a x to the power of 2 end exponent plus 2 b x plus c over denominator A x to the power of 2 end exponent plus 2 B x plus C end fraction$ gives only one value of $x$

This is possible only when $a x to the power of 2 end exponent plus 2 b x plus c minus y open parentheses A x to the power of 2 end exponent plus 2 B x plus C close parentheses$ is a perfect square

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