If $q>p>0$ show that the point $(x,y)=(0,0)$ minimizes the function $$f(x,y) = (y-px^2)(y-qx^2)$$ locally on the lines $$y=ht, x=kt$$ through the point (x,y).
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
Just insert the definition of $y$ and $x$ so define the function $$g(t)=f(kt,ht)=\ldots$$ Now you have a function $g:R\rightarrow R$, where you can use standard one-dimensional analysis to show that $t=0$ is a minimal point (under certain conditions on $q$ and $p$ maybe). But this means that $f$ has a minimum in $(0,0)$ if you only consider sequences in $R^2$ that are linear (you walk on lines into $(0,0)$.
You did not ask this, but the natural question would be now: Is $(0,0)$ a minimal point of $f$.
Is this helpful?