Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$ . Then f has a relative maximum at

A) $\left( \frac { \pi } { 5 } , \frac { \pi } { 5 } \right)$
B) $\left( - \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$
C) $\left( \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$
D) $\left( - \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$
E) $\left( \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$

Question

Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$ . Then f has a relative maximum at

A) $\left( \frac { \pi } { 5 } , \frac { \pi } { 5 } \right)$
B) $\left( - \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$
C) $\left( \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$
D) $\left( - \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$
E) $\left( \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$

Quizplus · Accepted Answer

To find the relative maximum of $f(x, y) = \sin(x + y) + \sin x + \sin y$, we first find its partial derivatives with respect to $x$ and $y$, and then set them to zero to find critical points. The partial derivative with respect to $x$ is $f_x = \cos(x + y) + \cos x$, and with respect to $y$ is $f_y = \cos(x + y) + \cos y$. Setting $f_x = 0$ and $f_y = 0$ gives us equations that are satisfied by $\left( \frac{\pi}{3}, \frac{\pi}{3} \right)$ among the given options. To confirm it's a maximum, we would check the second partial derivatives and use the second derivative test, but given the options and the nature of the sine function, $\left( \frac{\pi}{3}, \frac{\pi}{3} \right)$ is the correct answer for a relative maximum.

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Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$