Suppose you have a region $R$ that satisfies $4x^2+y^2\leq 4$ What is the absolute maximum of the function $f(x,y) = 4x^2 + 5y$ on $R$?
The correct answer is $10$
First, I calculated the critical points in $R$:
$$f_x = 8x$$
$$8x= 0$$
$$x=0$$
$$f_y = 5$$
$$5 \ne 0$$
Given the fact that there are no points where both $f_x$ and $f_y$ equal $0$, doesn't that mean there are no critical points in $R$, indicating no maximum value? I don't know if I have to check the boundary in this case, since the region is circular? I am not allowed to use Lagrange Multipliers.
Any insight into how to solve these types of problems would help.
Best Answer
The function $f(x)=x$ also has an absolute maximum on $[-1,1]$, even though there is no point at which $f'(x)=0$.
Remember, the zeroes of the partial derivatives only give you candidates for the critical points in the interior of the domain of $f$. On the border (where a maximum can also be reached sometimes), you can use Lagrange multipliers to calculate the max.
Alternatively, your particular function is fairly simple, so you can do without Lagrange. You have, on the border, $4x^2+y^2=4$, which means you can express $x^2$ as a funciton of $y$, and insert that into $f(x,y)$ to get a function of only $y$, and you can calculate the maximum of $f$ on the border quite easily.