List the critical points of the function $f ( x , y ) = ( x - 9 ) ^ { 5 } ( y + 5 ) ^ { 3 } \text { for which the }$ Second Partial Test fails.
A) Test fails at $( - 9 , a )$ and $( b , 5 )$.
B) Test fails only at $( 9 , - 5 )$.
C) Test fails at $( 9 , a )$ and $( b , - 5 )$.
D) Test fails at $( 9 , a )$ and $( b , 5 )$.
E) Test fails only at $( 9,5 )$.

Question

List the critical points of the function $f ( x , y ) = ( x - 9 ) ^ { 5 } ( y + 5 ) ^ { 3 } \text { for which the }$ Second Partial Test fails. 
A) Test fails at $( - 9 , a )$ and $( b , 5 )$.
B) Test fails only at $( 9 , - 5 )$.
C) Test fails at $( 9 , a )$ and $( b , - 5 )$.
D) Test fails at $( 9 , a )$ and $( b , 5 )$.
E) Test fails only at $( 9,5 )$.

Quizplus · Accepted Answer

The critical points of the function are where the first partial derivatives with respect to x and y are zero. For $f(x, y) = (x - 9)^5(y + 5)^3$, the first partial derivatives are $f_x = 5(x - 9)^4(y + 5)^3$ and $f_y = 3(x - 9)^5(y + 5)^2$. Setting these equal to zero, we find that $f_x = 0$ when $x = 9$ for any y, and $f_y = 0$ when $y = -5$ for any x. Therefore, the critical points are of the form $(9, a)$ and $(b, -5)$, where $a$ and $b$ can be any real number. The Second Partial Derivative Test fails to give information at these points because the function's behavior depends on higher-order derivatives at these points due to the nature of the exponents in the given function.

List the Critical Points of the Function f(x,y)=(x−9)5(y+5)3 for which the f ( x , y ) = ( x - 9 ) ^ { 5 } ( y + 5 ) ^ { 3 } \text { for which the }f(x,y)=(x−9)5(y+5)3 for which the

List the Critical Points of the Function $f ( x , y ) = ( x - 9 ) ^ { 5 } ( y + 5 ) ^ { 3 } \text { for which the }$