Show that $\displaystyle z=(x+y)^2$ has infinity many critical points.
I've got the formula of the tangent plane to be $\displaystyle z=(a+b)(2x+y2-a-b)$ where a and b are the points they touch the surface, so they too are variables.
Show that $\displaystyle z=(x+y)^2$ has infinity many critical points.
I've got the formula of the tangent plane to be $\displaystyle z=(a+b)(2x+y2-a-b)$ where a and b are the points they touch the surface, so they too are variables.
If you set the partial derivatives equal to zero, as a system, you should see that there are infinitely many solutions.
$\displaystyle \frac{dz}{dx} = 2(x+y) = 0$
$\displaystyle \frac{dz}{dy} = 2(x+y) = 0$
It's the set of all points 3-dimensional space such at x=-y.
I hope this helps.