Is this function coercive? Is my method reasonable?

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Is this function coercive?

$$f(x) = (x_1+2x_2)^2$$

I thought that because, for $x=(t, -t/2)'$ (whose norm goes to infinity when t does), $f(x)=0$, that this would mean that $f$ is not coercive, but I'm getting conflicting answers via different methods. Does anyone have any insights? Is this method valid?

Thank you!