Show that $f$ has a stationary point at $(0,0)$ for every $k ∈ R$. $f(x,y) = x^2 + kxy + y^2$ I know that the definition of a stationary point is when the gradient is $0$ so local maximum or local min or point of inflection. But unsure how to show that the point is stationary at $(0,0)$ for every $k ∈ R$.
Show that $f$ has a stationary point at $(0, 0)$ for every $k ∈ R$.
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calculus
multivariable-calculus
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1Points of inflection are not necessarily stationary points. Consider sine. – 2017-01-03
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$\nabla f = (f_x,f_y) = (2x+ky,2y+kx) = (0,0)$ at $(x,y) = (0,0)$. This concludes the claim.
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0it is concluded because $2(0)+k(0)=0$ and $ 2(0)+k(0)=0$ correct? – 2017-01-03
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0Yes, you are correct. – 2017-01-03