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I was just wondering how to answer this kind of question:

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How do I answer this question using concepts such as stationary point, directional derivative, and linear independence?

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Brief outline: The vectors (1, 1) and (1, -1) are linearly independent in $\mathbb{R}^2$, then the gradient of $f$ at $x^*$ is equal to $0$, and hence $x^*$ is a stationary point of $f$. Try and fill in the details yourself!

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  • $\begingroup$ Hi, Thank you very much. How do you know that the derivative of f at x* is equal to 0 though? At what point is it equal to 0? Also, when you say derivative, you mean gradient or directional derivative? $\endgroup$ – Jun Jang Sep 23 '17 at 17:54
  • $\begingroup$ I meant gradient, I'll amend that to clarify. $\endgroup$ – B. Mehta Sep 23 '17 at 17:55
  • $\begingroup$ The gradient of $f$ at $x^*$ is zero, which you should be able to prove using ideas from multivariable calculus $\endgroup$ – B. Mehta Sep 23 '17 at 17:57
  • $\begingroup$ i was wondering if you could show how the fact that 'v1, v2,.... vn are linearly independent vectors' implies that the gradient of f at x* is equal to 0. $\endgroup$ – Jun Jang Sep 24 '17 at 17:01

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