This probably can be done with some clever manipulation of inequalities but the problem does suggest that the method of Lagrange multiplier would be effective.
Thanks for participating, Jester and well done! And now it seems to me Lagrange Multiplier is a magic tool and can be applied to all kind of optimization problems to find the desired extrema points.
I want to share with you and others the solution proposed by other as well. It suggests the use of Hölder's Inequality to solve this problem.
If we let \(\displaystyle x=\left(\frac{a^3}{c} \right)^{\frac{2}{3}}\) and \(\displaystyle y=\left(\frac{b^3}{d} \right)^{\frac{2}{3}}\), we see that