Consider the function $f:\mathbb{R}P^2 \to \mathbb{R}$ given by
$$[x^1,x^2,x^3] \to \frac{3(x^1)^2}{(x^1)^2+(x^2)^2+(x^3)^2}$$.
- Find all the critical points of $f$. Hint: you need to check in all three of our standard coordinate charts $\{(Ui,\phi_i)\}_{i=1,2,3}$.
- $(U_1,\phi_1)$:
$Id \circ f \circ (\phi_1)^{-1} (x^1,x^2)=f (1,x^1,x^2)= \frac{3}{1+(x^1)^2+(x^2)^2}$
We have $$[f_*]= \begin{bmatrix} \frac{-6x^1}{(1+(x^1)^2+(x^2)^2)^2} & \frac{-6x^2}{(1+(x^2)^2+(x^2)^2)^2} & 0 \end{bmatrix}$$
This equals to $0$ when $x^1=x^2=0$. So the critical points take the form $(0,0,a)$ where $a\in \mathbb{R}$
- $(U_2,\phi_2)$:
$Id \circ f \circ (\phi_2)^{-1} (x^1,x^2)=f(x^1,1,x^2)=\frac{3(x^1)^2}{1+(x^1)^2+(x^2)^2}$
We have $$[f_*]= \begin{bmatrix} \frac{6x^1(1+(x^2)^2)}{(1+(x^1)^2+(x^2)^2)^2} & \frac{-6(x^1)^2x^2}{(1+(x^1)^2+(x^2)^2)^2} & 0 \end{bmatrix}$$
This equals to $0$ when $x^1=0$. So the critical points take the form $(0,b,c)$ where $b,c \in \mathbb{R}$.
- For $(U_3,\phi_3)$, I got the same as the chart $(U_2,\phi_2)$.
As a result, the critical points $\{(0,0,a),(0,b,c)\}$.
Is it correct?
- Find the point at which f takes its maximum value.
I know that I should substitute the critical points into $f$. But I got $0$ in all of them:
$f(0,0,a)=0$
$f(0,b,c)=0$
My teacher told me that only one of them would be $0$, so I think there is something wrong in the step $1$.
Any help would be appreciated. Thanks