Indifference curve can be convex or concave. Convex indifference curve is the one you first talk about, and what we usually see. Concave is the opposite. So how the indifference curve looks like is determined by utility function. If a function, say f(x), f'(x)>0. f"(x)<0, it's convex. If f"(x)>0, it's concave.
To solve for the optimal point (or the point with highest utility), you need to have a utility function and a budget constraint. K=MUx/MUy is also called Marginal Rate of Substitution. If the budget constraint is Px*X+Py*P=I, set K=Px/Py can give you the solution. So from the graph, the higher the indifference curve, the higher the utility level. If the curve is convex, the highest point must be a tangency point with the budget constraint line. If the curve is concave, the highest curve that touches the budget constraint must be either on horizontal axis or vertical axis, and we call that "corner solution".
简单说，无差别曲线可以是向原点凸，也可以向原点凹， 取决于utility function. 正常的商品大部分是消费的越多，边际效益递减，所以是向原点凸的。无论是什么形状，都是越往右上方移，utility越高。 budget constraint 可以画成一条直线，负斜率的那种。算MUx/MUy=Px/Py=K, 就是在找他们相切的点。所以如果是向原点凸的无差别曲线，最优点一定是在 budget constraint的直线和无差别曲线相切的点。 因为那个是可以达到的最高的一条无差别曲线。 如果是向原点凹的无差别曲线，那么切点并不是最优点。因为还可以把曲线再往右上方移，直到无差别曲线横轴或者纵轴相交的点刚好重合在 budget constraint上，那样才是最优。
不知道我说清楚没有，我在美国学的这些，所以有些词也翻译不好，自己明白也不一定讲的明白，呵呵。总之画画图就一目了然了。
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