Determine the optimal solution to the following constrained maximization. Then consider the level curve of F passing through the optimal solution poin and show that its slope at the point is equal to the slope of the budget line $3x+y=b$.
I have tried to determine the optimal solution and I got $x=\frac{b}{7}$ and $y=\frac{4b}{7}$. However the slopes of the level curve and the budget line don't corrispond (I get $\frac{-3}{2}$ and $-3$) so I guess I made a mistake at some point.
Let $x=m^2,\;m≥0$ and $y=3n^2,\; n≥0$ and denote that $A=F(x,y)$ , then you have:
$$ \begin{align}&\begin{cases} m+2n=A,\;A≥0\\3m^2+3n^2=b\end{cases}\\ \implies &(A-2n)^2+n^2-\frac b3=0\\ \implies &A^2-4An+5n^2-\frac b3=0\\ \implies &5n^2-4An+\left(A^2-\frac b3\right)=0\\ \implies &5\left(n-\frac {2A}{5}\right)^2=\frac b3-\frac {A^2}{5}≥0\\ \implies &\frac {A^2}{5}-\frac b3≤0\\ \implies &A≤\sqrt{\frac {5b}{3}}\end{align} $$
If $A=\sqrt{\dfrac {5b}{3}}$, then $n=\dfrac {2A}{5}$. Thus, you can determine the pair of $x,y$ from the last result.