How can you maximize your happiness under a budget? When does a function reach its minimum value? When does a curve change direction? The calculus of extrema explains these "extreme" situations.

Let \(\alpha \)=\(\frac { a }{ b }\). Where 'a' and 'b' are integers.

Then find the value of \(\quad |a\times b|\).

**DETAILS AND ASSUMPTIONS**

\(\bullet \) Assume that Mathematician is moving in Co-ordinate X-Y plane.

\(\bullet \) There are infinite number of Shops at Co-ordinate X-axis.So Mathematician(Romeo) choose Particular Shop To impress his Girlfriend(Juliet).

\(\bullet \) Treat all object as point object.

\(\bullet\) May be Fermat's Principle of Light waves Helpful Here .

As the radius of circle \(X\) increases, at what radius of circle \(X\) does the sum of the areas of circle \(X\) and \(Y\) change the slowest?

Express that radius as a fraction of the side of the square \(ABCD\), and round to the nearest hundredth.

\[16y^2=a+15 x-9 x^2+x^3\]

Note: \(b\) is a constant of real integral value.

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