So given a function $$f(x,y)=x^3+30xy+2y^3,$$ I am able to find the maximum value in a specific region. However, I need to be able to calculate the distance that would be traversed by a particle moving between the origin and the maximum, as if it were hypothetically climbing and descending the function.
Basically I need to find the arc length of the function, along the path between the origin and the maximum point. How would I go about doing this?
Any help would be appreciated
If you're not interested in any particular path on the surface joining $(0,0,f(0,0))$ to the point on the surface where $f$ achieves its maximum value, then let's just consider a straight line in the plane that joins $(0,0)$ to $(a,b)$, where I'm assuming that it's at $(a,b)$ where $f$ achieves its maximum value. Here's our line, parametrically: $c(t):=(at,bt)$ for $0\leq t\leq 1$. Then $$\tilde{c}(t):=(at,bt, (at)^{3}+30abt^{2}+2(bt)^{3})$$ is a path on the graph of $f$ that joins $(0,0,f(0,0))$ to the highest point on the graph. Find the length of this curve by computing the following, using standard calculus $1$ methods (you may or may not need trig substitution; I haven't worked it out that far): $$\int_{0}^{1}||\tilde{c}'(t)||dt=\int_{0}^{1}\sqrt{a^{2}+b^{2}+[3(at)^{2}+60abt+6(bt)^{2}]^{2}}dt$$