Find the absolute maximum and minimum values of the function on the given curve.
-Function: $f ( x , y ) = x + y ;$ curve: $x ^ { 2 } + y ^ { 2 } = 49 , y \geq 0$. (Use the parametric equations $x = 7 \cos t , y = 7 \sin t$.)
A) Absolute maximum: $7 \sqrt { 2 }$ at $t = \frac { \pi } { 4 }$; absolute minimum: $- 7 \sqrt { 2 }$ at $t = \frac { 3 \pi } { 4 }$
B) Absolute maximum: $7 \sqrt { 2 }$ at $t = \frac { \pi } { 4 }$; absolute minimum: $- 7$ at $t = \pi$
C) Absolute maximum: 7 at $t = \frac { \pi } { 2 } ;$ absolute minimum: $- 7 \sqrt { 2 }$ at $t = \frac { 3 \pi } { 4 }$
D) Absolute maximum: 7 at $t = \frac { \pi } { 2 }$; absolute minimum: $- 7$ at $t = \pi$

Question

Quizplus · Accepted Answer

Given the function $f(x, y) = x + y$ and the curve $x^2 + y^2 = 49$ with $y \geq 0$, using the parametric equations $x = 7\cos t, y = 7\sin t$, we substitute these into the function to get $f(t) = 7\cos t + 7\sin t$. To find the maximum and minimum values, we can analyze this function or directly observe that the maximum value occurs when both $\sin t$ and $\cos t$ are positive and equal, which happens at $t = \frac{\pi}{4}$, giving a maximum value of $7\sqrt{2}$. The minimum value occurs when $t = \pi$, where $\cos t = -1$ and $\sin t = 0$, giving a minimum value of $-7$. This accounts for the constraint $y \geq 0$, limiting $t$ to the range $[0, \pi]$.

Find the Absolute Maximum and Minimum Values of the Function f(x,y)=x+y;f ( x , y ) = x + y ;f(x,y)=x+y;

Find the Absolute Maximum and Minimum Values of the Function $f ( x , y ) = x + y ;$