Question
If $$xy = {a^2}$$ and $$S = {b^2}x + {c^2}y$$ where $$a,\,b$$ and $$c$$ are constants then the minimum value of $$S$$ is :
A.
$$abc$$
B.
$$bc\sqrt a $$
C.
$$2abc$$
D.
none of these
Answer :
$$2abc$$
Solution :
$$\eqalign{
& S = {b^2}x + \frac{{{c^2}{a^2}}}{x} \cr
& \therefore \frac{{dS}}{{dx}} = {b^2} - \frac{{{c^2}{a^2}}}{{{x^2}}} = 0 \cr
& \Rightarrow x = \pm \frac{{ca}}{b} \cr
& \frac{{{d^2}S}}{{d{x^2}}} = 2{c^2}{a^2}.\frac{1}{{{x^3}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\therefore {\left. {\frac{{{d^2}S}}{{d{x^2}}}} \right)_{x = \frac{{ca}}{b}}} > 0 \cr
& \therefore \,\,\min \,S = {b^2}.\frac{{ca}}{b} + \frac{{{c^2}{a^2}}}{{\frac{{ca}}{b}}} = 2abc \cr} $$