The book used lagrange multipliers to solve this. Here's the solution as per book. $$ F = xyz + k(x+y+z-a)$$ Solved to get $$(x,y,z) = (\frac{a}{3},\frac{a}{3},\frac{a}{3})$$
Now,Verifying whether this point corresponds to maxima or minima but $$F_{xx} = 0 $$.
As it can't be verified using second derivatives of $F$ , the book considers $z$ as a function of $x$ and $y$ and verifies by second derivatives of $z$.
My questions are
-When can we assume $z$ as a function of $x$ and $y$, even when it is given in the problem as independent variable?
-We have to find extreme values of $F$ in the question. Then how can second derivatives test of $z$ verify the answer?