## [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.