$$a$$ , $$b$$ are real numbers such that $$2a^2+5b^2=20$$, then the maximum value of $$a^4b^6$$ is

Let $$u=a^2$$ and $$v=b^2$$, so $$2u+5v=20$$ and the expression is $$u^2v^3$$. The maximum occurs when the two resource terms are divided in the ratio of the exponents, so $$2u=8$$ and $$5v=12$$. Hence $$u=4$$, $$v=\frac{12}{5}$$ and the maximum is $$16\left(\frac{12}{5}\right)^3=\frac{27648}{125}=221.184$$. This value is not among the printed options, so the source treats the problem as a bonus and the answer letter is only a required placeholder.