Find the absolute extrema of the function on the closed triangular region with vertices , and .
A)Absolute minimum 5, Absolute maximum 17
B)Absolute minimum -5, Absolute maximum 5
C)Absolute minimum 0, Absolute maximum 5
D)Absolute minimum -5, Absolute maximum 17

Question

Find the absolute extrema of the function  on the closed triangular region with vertices  ,  and  .
A)Absolute minimum 5, Absolute maximum 17 
B)Absolute minimum -5, Absolute maximum 5 
C)Absolute minimum 0, Absolute maximum 5 
D)Absolute minimum -5, Absolute maximum 17

Quizplus · Accepted Answer

To find the absolute extrema, we need to evaluate the function at all critical points and at the vertices of the region. Since the region is a triangle, we have three vertices to consider. Let's first find the critical points (points where the gradient is zero or the boundary of the region). Calculating the gradient of the function, ∇f(a,b) = <6a, 6b - 6, 12a> ∇f(a,b) = <0, 0, 0> when a=0 and b=1 Now, we need to evaluate the function at these critical points and at the vertices of the region: f(11eaa3e5_54de_b126_9f8f_417171c34506_TB5971_11) = 11, f(11eaa3e5_54de_b127_9f8f_6f9e01d77f08_TB5971_11) = 5, f(11eaa3e5_54de_b128_9f8f_fd2fb40ad238_TB5971_11) = 17, f(0,1) = 5; So, the absolute minimum is 5 at (0,1) and the absolute maximum is 17 at (11eaa3e5_54de_b128_9f8f_fd2fb40ad238_TB5971_11). Therefore, the answer is (D).

Find the Absolute Extrema of the Function on the Closed