Question

Find the absolute maxima and minima of the function on the given domain. T(x,y)=x2+xy+y2−12x+7 on the rectangular plate 0≤x≤9,−5≤y≤0

The absolute maximum occurs at_______________

Answer 1

To find the absolute maxima and minima of a function, we need to find the critical points by taking the partial derivatives and setting them equal to zero. Then, we need to check the endpoints of the given domain and compare the values to find the absolute maximum and minimum. In this case, the absolute maximum occurs at T(9,-5) = 53.

Step-by-step explanation:

To find the absolute maxima and minima of the function T(x,y)=x^2+xy+y^2−12x+7 on the given domain, we first need to find the critical points by taking the partial derivatives and setting them equal to zero.

∂T/∂x = 2x+y-12 = 0

∂T/∂y = x+2y = 0

Solving these two equations simultaneously, we get x = 8 and y = -4.

Next, we need to check the endpoints of the rectangular plate. Plugging in x = 0, y = 0 gives T(0,0) = 7. Plugging in x = 9, y = -5 gives T(9,-5) = 53.

Comparing the values at the critical points and endpoints, we find that the absolute maximum occurs at T(9,-5) = 53.