Find the extreme values of the function subject to the given constraint.
-$f ( x , y ) = 7 x ^ { 2 } + 3 y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = 1$
A) Minimum: 3 at $( 0 , \pm 1 )$; maximum: 0 at $( 0,0 )$
B) Minimum: 3 at $( \pm 1,0 )$; maximum: 7 at $( 0 , \pm 1 )$
C) Minimum: 3 at $( 0 , \pm 1 )$; maximum: 7 at $( \pm 1,0 )$
D) Minimum: 3 at $( \pm 1,0 )$; maximum: 0 at $( 0,0 )$

Question

Quizplus · Accepted Answer

We use Lagrange Multipliers to solve the problem:
Let $g(x,y) = x^2 + y^2 - 1$ be the constraint.
Then, we need to solve the system of equations:
\begin{align*}

abla f &= \lambda 
abla g \
g(x,y) &= 0
\end{align*}
That is,
\begin{align*}
14x &= 2\lambda x \
6y &= 2\lambda y \
x^2 + y^2 &= 1
\end{align*}
Simplifying the first two equations, we get $\lambda = 7$ and $\lambda = 3$, respectively. Since $\lambda$ can't take on two different values, we must have $y = 0$ (and then $x = \pm 1$) in the second equation, which gives us the minimum value of $f$, that is, $3$. Substituting these values into the third equation, we can see that the points $(0,1)$ and $(0,-1)$ are the candidates for the maximum, and plugging them into $f$ we get $7$ in both cases. Therefore, the maximum value of $f$ is $7$ and the minimum value of $f$ is $3$. The only choice that has this as an option is C.

Find the Extreme Values of the Function Subject to the Given