So this was the problem I was trying to solve, and I got stuck in part (b).
When you total differentiate $U(x,y)=0$, we can see that $Uxdx+Uydy=0$, which means $dy/dx=-Ux/Uy$
However, I am struggling with the geometric/mathematical intuition part. When $dU=0$, it is telling us that it will be a stationary point of U. Then looking at $dy/dx=-Ux/Uy$, I first thought when $x$ and $y$ changes, they change so that $Ux$ and $Uy$ will change in opposite signs/directions, thereby causing the whole utility to decrease, making the point a maximum. However as I think anout it, I noticed that this interpretation wouldn’t work for the minimum stationary point. Can someone help me find out what I am thinking wrong?
Too long for a comment:
Your notation looks wrong. By $Ux,Uy$ you must mean the partial derivatives of $U$ which are better denoted by $U_x,U_y\,.$
It is not correct that $dU=0$ characterizes the extrema of $U\,.$ It characterizes the level sets of $U\,,$ i.e., $U=$ const.
What relationship of the gradient of $U$ and the curves in the $(x,y)$-plane that follow such level sets?
Hint: use the implicit function theorem to find $y(x)$ from $U(x,y)=$ const. Then parametrize the level set by $x\mapsto (x,y(x))$ and use the chain rule on $\frac{d}{dx}U(x,y(x))\,.$