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Let $\displaystyle \phi(x,y,z)=x^{2}+4y^{2}-2yz-z^{2}$ and let $\displaystyle x_{0}=2e_{1}+e_{2}-4e_{3}$. So i have to verify the hypotheses of the implicit function theorem for the above example. I am done in following manner: $$\phi(x_{0})=2^{2}+4 (1)^{2}-2(1)(-4)-(-4)^{2}=4+4+8-16=0$$ Then since $\displaystyle d\phi(x,y,z)=[2x,8y-2z,-2y-2z]$ I have $\displaystyle d\phi(2,1,-4)=[4,16,6]$ After this i dont know what i have to do? Please suggest me!

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