Find values of x,y,and ? that satisfy the system.These systems arise in certain optimization problems in calculus,and ? is called a Lagrange multiplier.? $$\left\{ \begin{array} { c }
5 x - 5 x \lambda = 0 \
- 2 y + \lambda = 0 \
y - x ^ { 2 } = 0
\end{array} \right.$$ ?
A)? $x = \pm \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = - 1 \text { or } x = 0 , y = 0 , \lambda = 0$
B)? $x = \pm \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 1 , \lambda = 1$
C)? $x = \pm \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 0 , \lambda = 0$
D)? $x = \pm \frac { \sqrt { 2 } } { 2 } , y = - \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 0 , \lambda = 0$
E)? $x = - \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 0 , \lambda = 0$

Question

Find values of x,y,and ? that satisfy the system.These systems arise in certain optimization problems in calculus,and ? is called a Lagrange multiplier.? $$\left\{ \begin{array} { c } 
5 x - 5 x \lambda = 0 \
- 2 y + \lambda = 0 \
y - x ^ { 2 } = 0
\end{array} \right.$$ ?
A)? $x = \pm \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = - 1 \text { or } x = 0 , y = 0 , \lambda = 0$
B)? $x = \pm \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 1 , \lambda = 1$
C)? $x = \pm \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 0 , \lambda = 0$
D)? $x = \pm \frac { \sqrt { 2 } } { 2 } , y = - \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 0 , \lambda = 0$
E)? $x = - \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 0 , \lambda = 0$

Quizplus · Accepted Answer

From the first equation, $5x - 5x\lambda = 0$, we can factor out $5x$ to get $5x(1-\lambda) = 0$. This gives us two possibilities: $x = 0$ or $\lambda = 1$. From the second equation, $-2y + \lambda = 0$, if $\lambda = 1$, then $y = \frac{1}{2}$. Substituting $y = x^2$ from the third equation, we get $x^2 = \frac{1}{2}$, which gives $x = \pm \frac{\sqrt{2}}{2}$. If $x = 0$, from the third equation, $y = 0$, and from the second equation, $\lambda = 0$. Therefore, the solutions are $x = \pm \frac{\sqrt{2}}{2}, y = \frac{1}{2}, \lambda = 1$ or $x = 0, y = 0, \lambda = 0$.

Find Values of X,y,and ? That Satisfy the System {5x−5xλ=0−2y+λ=0y−x2=0\left\{ \begin{array} { c } 5 x - 5 x \lambda = 0 \\- 2 y + \lambda = 0 \\y - x ^ { 2 } = 0\end{array} \right.⎩⎨⎧​5x−5xλ=0−2y+λ=0y−x2=0​

Find Values of X,y,and ? That Satisfy the System $\left\{ \begin{array} { c } 5 x - 5 x \lambda = 0 \\- 2 y + \lambda = 0 \\y - x ^ { 2 } = 0\end{array} \right.$