2
$\begingroup$

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 $d=(-2,1)^T$

$\endgroup$
  • 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
  • $\begingroup$ @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
1
$\begingroup$

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.