Find a unit normed vector (direction) $d$ that minimizes the directional derivative, where $\nabla f(x^*)=(12,24)^T$.

My question: Is the directional derivative minimized by selecting a vector orthogonal to the gradient? Is so would $d=(-2,1)^T$

  • 2
    $\begingroup$ A derivative can also be negative. $\endgroup$
    – user251257
    Feb 10 '18 at 22:47
  • $\begingroup$ what do you mean? $\endgroup$
    – thisisme
    Feb 10 '18 at 22:49
  • 1
    $\begingroup$ @thisisme Please, if you are ok, you can accept the answer and set it as solved. Thanks! cdn.sstatic.net/img/faq/faq-accept-answer.png $\endgroup$
    – user
    Feb 12 '18 at 14:07
  • $\begingroup$ thank you for your help! by marking the green arrow does it consider to be solved? @gimusi $\endgroup$
    – thisisme
    Feb 12 '18 at 23:57

The directional derivative, for every direction defined by a vector $\vec v$, is given by the doct product with the gradient, that is:

$$f_{\vec v}=\frac {\partial f}{\partial \vec v}=\nabla f \cdot \vec v$$

which is

  • max (positive) when $\vec v$ is in the direction (i.e. positive multiple) of $\nabla f$
  • zero when $\vec v$ is orthogonal to $\nabla f$
  • min(negative) when $\vec v$ is in opposite direction with respect to $\nabla f$

thus the unit vector which minimize the directional derivative is

$$u=\left(-\frac1{\sqrt5},-\frac2{\sqrt5} \right)$$


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