min\max F(x,y)=3xy-4x^2-y^2+150x+50 Constraints: 1)x>=y^2 2)x>=3^2 What is the function for finding suspicious points?

To find the critical or suspicious points for the function F(x, y) = 3xy - 4x^2 - y^2 + 150x + 50 with constraints x \geq y^2 and x \geq 9 , follow these steps:

### 1. Calculate the Partial Derivatives:
F_x = \frac{\partial F}{\partial x} = 3y - 8x + 150
F_y = \frac{\partial F}{\partial y} = 3x - 2y

### 2. Set the Partial Derivatives to Zero:
Set F_x = 0 and F_y = 0 :
3y - 8x + 150 = 0
3x - 2y = 0

### 3. Solve the System of Equations:
From 3x - 2y = 0 , we get y = \frac{3}{2}x . Substitute this into the first equation:
\frac{9}{2}x - 8x + 150 = 0
-\frac{7}{2}x + 150 = 0
x = \frac{150}{\frac{7}{2}} = \frac{300}{7} \approx 42.86
Substitute x back to find y :
y = \frac{3}{2} \times \frac{300}{7} \approx 64.29

### 4. Check Constraints:
Evaluate:
x \geq y^2 \Rightarrow \frac{300}{7} \geq \left(\frac{3}{2} \times \frac{300}{7}\right)^2
x \geq 9 \Rightarrow \frac{300}{7} \geq 9

### Conclusion:
The critical point is at \left(\frac{300}{7}, \frac{450}{7}\right) .

\boxed{(\frac{300}{7}, \frac{450}{7})} is the critical point.