Let $f(x,y)$ be a differentiable function.
Suppose that for every line, $l$ , that passes through $(0,0)$ the point $(0,0)$ is a local minimum of $f(x,y)\mid_ l$. Does that mean that $(0,0)$ is a local minimum of $f(x,y)$?
When I first saw the question I tried to prove it by letting $l=(a,b)\cdot t$ where a,b are constants. Then I got $x=at, y=bt$ and substituted that in the function $f(x,y)$ to get $f(at,bt)$. Then I used the chain rule to differentiate $f$ , but didn't know what to do from there.
Is this a good direction? or am I suppose to disprove the claim?
Any help will be appreciated!