I've asked a very similar question to this before but a few days later I'm looking at another question and am still slightly confused.
I have the function: $f(x,y) = x/(x+y)$
I want to the find the direction in which the directional derivative is $0$ at the point $(1,2)$.
I started by calculating my partial derivatives (in order to obtain the gradient) and I got:
$f_x = y/(x+y)^2$
$f_y = -x/(x+y)^2$
So the gradient of $f$ is $(y/(x+y)^2 , -x/(x+y)^2)$
I plugged in $(1,2)$ to get the direction where the directional derivative is maximal (it asks this earlier on in the question) and got $(2/9 , -1/9)$
In my previous post on here someone helped me before and told me that the directional derivative will be $0$ when perpendicular to the gradient, so I tried this. However, I either found the perpendicular incorrectly (which is probably what's wrong here) or the method doesn't work here.
I found the perpendicular to be $\pm (x/(x+y)^2 , y/(x+y)^2)$ but when I plug my $x$ and $y$ values into this I don't get the correct answer (I have the solutions sheet only, no working out given, and the solutions should be $(1,2)$ and $(-1,-2)$).
If anyone could show me what I'm doing wrong and the method for calculating the perpendiculars (if needed) it would be really helpful.
Thanks in advance
(The old question, for reference: Find the direction where the rate of change of the function is 0 )
Best Answer
The directional derivative is indeed perpendicular to the gradient.
Since the gradient of your function is $\left(\frac29,-\frac19\right)$ at $(1,2),$ you can rotate that vector by a right angle in either direction to obtain either $\left(\frac19,\frac29\right)$ or $\left(-\frac19,-\frac29\right).$
But if a vector $v$ points in a particular direction, so does $kv$ for any positive scalar multiplier $k$. If you multiply each of the vectors $\left(\frac19,\frac29\right)$ and $\left(-\frac19,-\frac29\right)$ by the scalar $9$ you get $(1,2)$ and $(-1,-2).$
By the same token you can multiply $(1,2)$ and $(-1,-2)$ each by $\frac19$ and get the vectors $\left(\frac19,\frac29\right)$ and $\left(-\frac19,-\frac29\right),$ which point in the same directions as $(1,2)$ and $(-1,-2).$
Unless there are additional restrictions on the answer, such as a requirement that the magnitude by $\sqrt5$ or that the magnitude of the first component by $1,$ I don't see how $(1,2)$ and $(-1,-2)$ can be said to be the only correct ways to write the answer.
On the other hand, the only correct answers are the direction of $(1,2)$ and the direction of $(-1,-2),$ expressed in some suitable ways (for example, expressed as the directions of $\left(\frac19,\frac29\right)$ and $\left(-\frac19,-\frac29\right)$). Is it possible that that is how the answer key was intended to be interpreted?