An algebra problem by Aly Ahmed

Algebra Level pending

2 x + 6 + 24 − 4 x \sqrt{2x+6} + \sqrt{24-4x}

Let M M and m m be the maximum value and the minimum value of the expression above for real x x , respectively.

Find M m \dfrac Mm .


The answer is 1.7320.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

The function f ( x ) = 2 x + 6 + 24 − 4 x f(x)=\sqrt {2x+6}+\sqrt {24-4x} is bitonic, possessing a local maximum at x = 0 , f ( 0 ) = 3 6 x=0,f(0)=3\sqrt 6 . The domain of definition of the function is [ − 3 , 6 ] [-3,6] . In this domain, the function has a minimum value at x = 6 , f ( 6 ) = 3 2 x=6,f(6)=3\sqrt 2 .

Therefore m = 3 2 , M = 3 6 m=3\sqrt 2,M=3\sqrt 6 , and M m = 3 ≈ 1.732 \dfrac{M}{m}=\sqrt 3\approx \boxed {1.732} .

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