The minimum value of $a^2+b^2+c^2+\frac1{a^2}+\frac1{b^2}+\frac1{c^2}?$ I came across the following problem :  

Let $a,b,c$ are non-zero real numbers .Then the minimum value of $a^2+b^2+c^2+\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}?$ This is a multiple choice question and the options are $0,6,3^2,6^2.$  

I do not know how to progress with the problem. Can someone point me in the right direction? Thanks in advance for your time.

By the symmetry of the expression, the minimum is attained with $a=b=c$ so we look for the minimum of 
$$f(x)=3(x^2+x^{-2})$$
By the derivative of the function $f$ we find that the minimum is at $x=1$, hence your answer must be $6$.
