Let $f(x, y) :=−3x^4−y^2+ 4x^2y$.
Prove that, for any $v∈R^2$, the function $h_v:R→R$ given by $h_v(t):=f(tv)$ has a local maximum at t= 0. Then find all local extrema of f, as well as $sup_{R^2} f$ and $inf_{R^2} f$.
To find the local maximum, i will have to find partial second order derivative for f(x,y) but is this the same as f(tv)? I know that If f has a local max or min at a then for h the function g(t)=f(a + th) has a local max or min at t = 0, but i dont know how to prove that.
In addition, how owuld i find all local extrema, sup and inf?
Thanks.
To show that $ h_v $ has a local maximum at $ 0 $ it is enough to calculate $ h_v'(0) $ and $ h_v''(0) $ (along with checking a special case). Next notice that a necessary condition for (a,b) to be a local extrema is that $$ \left. \frac{d}{dx} f(x,y) \right|_{(x,y)=(a,b)} = \left. \frac{d}{dy} f(x,y) \right|_{(x,y)=(a,b)} = 0. $$ This is enough to determine all local extrema. Finally to find sup and inf consider the functions $ x\mapsto f(x,0) $ and $ x\mapsto f(x,2x^2) $.