Nice use of AM GM

We are given 4 lines with equations:

• \(y = \Bigg( \dfrac{2 + 3a^2 + 3b^2(1 + a^2)^2 + b(6a + 6a^3)}{a + b(1 + a^2)} \Bigg)^2_{\min}\)
• \( y = \Bigg(\dfrac{c^2d^2 - 2c^2d + 2c^2 + 2cd - 2c + 1}{c^2d + c} \Bigg)_{\min} + 3 - \sqrt{8} \)
• \( y = [e]x^2 \)
• \( y =\dfrac{1}{2} [e]x^2 \)

With \( a,b,c,d > 0, e\geq 1\) .

The area bounded by the given lines and the curves when obtain a maximum value of \(M\), with \( M = \dfrac{a_{1}}{a_{2}}(\sqrt{a_{3}} - 1)(a_{4}\sqrt{a_{5}} - 1)\).

Find \( a_{1} + a_{2} + a_{3} + a_{4} + a_{5}\).

Details and Assumptions:

• \( [.] \) denote the greatest integer function.
• \( (f(x))_{\text{min}} \) denote the minimum value of \(f(x) \).

Dedicated to Aditya Raut and Guilherme Dela Corte.