Find the extreme values of the function subject to the given constraint.
-$f ( x , y ) = 3 x - y + 1 , \quad 3 x ^ { 2 } + y ^ { 2 } = 9$
A) Maximum: 7 at $\left( \frac { 3 } { 2 } , - \frac { 3 } { 2 } \right) ;$ minimum: $- 5$ at $\left( - \frac { 3 } { 2 } , \frac { 3 } { 2 } \right)$
B) Maximum: 4 at $\left( \frac { 3 } { 2 } , \frac { 3 } { 2 } \right) ;$ minimum: $- 2$ at $\left( - \frac { 3 } { 2 } , - \frac { 3 } { 2 } \right)$
C) Maximum: 4 at $\left( \frac { 3 } { 2 } , \frac { 3 } { 2 } \right) ;$ minimum: $- 5$ at $\left( - \frac { 3 } { 2 } , \frac { 3 } { 2 } \right)$
D) Maximum: 7 at $\left( \frac { 3 } { 2 } , - \frac { 3 } { 2 } \right) ;$ minimum: $- 2$ at $\left( - \frac { 3 } { 2 } , - \frac { 3 } { 2 } \right)$

Question

Quizplus · Accepted Answer

We can use the method of Lagrange multipliers to solve the problem. Let $g(x,y) = 3x^2 + y^2 = 9$ be the constraint function. Then, we consider the function $F(x,y,\lambda) = 3x - y + 1 - \lambda g(x,y)$. We want to find the critical points of $F$, which satisfy the following equations:

$$\begin{cases} \frac{\partial F}{\partial x} = 3 - 2\lambda x = 0 \ \frac{\partial F}{\partial y} = -1 - 2\lambda y = 0 \ \frac{\partial F}{\partial \lambda} = 3x^2 + y^2 - 9 = 0 \end{cases}$$

From the first equation, we have $x = \frac{3}{2\lambda}$. Substituting into the second equation, we obtain $y = -\frac{3}{2\lambda}$. Substituting both into the third equation, we get $\frac{27}{4\lambda^2} + \frac{9}{4\lambda^2} - 9 = 0$, which simplifies to $\lambda^2 = \frac{1}{4}$. Thus, we have two possible cases: $\lambda = \frac{1}{2}$ or $\lambda = -\frac{1}{2}$.

Case 1: $\lambda = \frac{1}{2}$
In this case, we have $x = 3$ and $y = -3$. Substituting into $f(x,y) = 3x - y + 1$, we obtain $f(3,-3) = 7$.

Case 2: $\lambda = -\frac{1}{2}$
In this case, we have $x = -3$ and $y = 3$. Substituting into $f(x,y) = 3x - y + 1$, we obtain $f(-3,3) = -5$.

Therefore, the maximum value of $f$ is $7$ and the minimum value of $f$ is $-5$, which agrees with choice (A).

Find the Extreme Values of the Function Subject to the Given