UPSC 2014 Maths Optional Paper 2 Q4b — Step-by-Step Solution

15 marks · Section A

Maxima and minima of multi-variable functions (analytic criteria) · Real Analysis · asked 5× in 14 yrs · Read the full method →

Question

Find the minimum value of $x^{2}+y^{2}+z^{2}$ subject to the condition $xyz=a^{3}$ by the method of Lagrange multipliers.

Standard 1-constraint Lagrange; symmetry of the objective (sum of squares) under sign flips and permutations.

Strategy. Standard Lagrange-multiplier setup; exploit symmetry to find critical points.

$L=x^{2}+y^{2}+z^{2}-\lambda(xyz-a^{3})$ .

\partial L/\partial x=2x-\lambda yz=0,\;\partial L/\partial y=2y-\lambda xz=0,\;\partial L/\partial z=2z-\lambda xy=0.

Multiply first equation by $x$ : $2x^{2}=\lambda xyz=\lambda a^{3}$ .

Similarly: $2y^{2}=\lambda a^{3}$ , $2z^{2}=\lambda a^{3}$ .

So $x^{2}=y^{2}=z^{2}$ .

$|x|=|y|=|z|$ . From $xyz=a^{3}$ (with $a>0$ assumed, so RHS positive), either:

In either case, $|x|=|y|=|z|=t$ for some $t>0$ , and $|xyz|=t^{3}=|a^{3}|=a^{3}$ , so $t=a$ .

$x^{2}+y^{2}+z^{2}=3t^{2}=3a^{2}$ .

\boxed{\;\min(x^{2}+y^{2}+z^{2})=3a^{2},\;\text{attained at}\;(a,a,a)\text{ (and other sign patterns).}\;}