Find Values of X, Y, And $\lambda$ That Satisfy the System $\lambda$

Multiple Choice

Find values of x, y, and $\lambda$ that satisfy the system.These systems arise in certain optimization problems in calculus, and $\lambda$ 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) $x = - \frac { \sqrt { 2 } } { 2 } , y = \frac { 1 } { 2 } , \lambda = 1 \text { or } x = 0 , y = 0 , \lambda = 0$