[Hints] 2014 Singapore MO (Junior Rd 1) Problem 32

2014 SMO (Junior-Rd1) 32. For a \geq \frac{1}{8}, we define

g(a) = \sqrt[3]{a + \displaystyle\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}} +\sqrt[3]{a - \displaystyle\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}.

Find the maximum value of g(a).

Hint 1: Could the expression under the cube root be rewritten as a cube?

Scroll down for Hint 2…

 

 

 

 

 

 

Hint 2: Guess that the expression under the cube root is of the form \left(\alpha + \sqrt{\frac{8a-1}{3}}\right)^3. Expand and compare terms with the actual expression under the cube root.

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