(Suggestion: in most of the problems that follow, you'll need to solve $\nabla f = \lambda \nabla g$. Rewrite this vector equation as several equations, then solve each equation for $\lambda$ before you substitute.)
- Find all points on the curve $g(x,y) = x^2y=16$ where the gradient of $f(x,y) = x^2+y^2$ is parallel to the gradient of $g(x,y)$.
- Find all points (in the first octant, so all positive coordinates) on the ellipsoid $g(x,y,z) = 4x^2+4y^2+z^2=4$ where the gradient of $f(x,y,z) = 8xyz$ is parallel to the gradient of $g(x,y,z)$.
- Find all points (again in the first octant) on the surface $g(x,y,z)=x^2+y^2+z=4$ where the gradient of $f(x,y,z)=4xyz$ is parallel to the gradient of $g(x,y,z)$
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