one solution to your first question.
obviously, one assumption your professor holds is that 'price is contant instead of a function of output'. Otherwise the hyperthesis is not correct. For example, consider P(x) = 1/x, F(x) = 4-X^2. Then Profit(x) = P(x)*F(x) - Wx = -x + 4 -Wx , which obviously is linear and not concave.

So we have P is constant. The proof will be very simple. Take first and secon derivative on Profit function = P*F(x) - W*x, you will find the second derivate of Profit is positively linear to the second derivative of F(x). So if F(x) is concave, Profit function is concave.

My answer might be wrong.

Regarding to your second question, I didn't quite get the question. It's unclear to me.

简言之吧:一种较为简单的方法就是看函数的二阶导数，如果函数的二阶导数为负,则该函数为凹函数(也称为下凸),反之,则为上凸函数.

你给出的生产函数F(x)是凹函数,即其F''（x）为负数, 利润函数 经计算你就知道它和F(x)的符号是相同的，即也为负,从而得证.不知道你们老师出这么一道题是想看你们的微积分知识还是想看你们的微观经济学知识.

 当然也有稍微比较麻烦点的方法，但也是可以很快就证明出来的.

[此贴子已经被作者于2007-11-7 6:25:17编辑过]