Chapter 13.9, Problem 1QCE

(a) Suppose that $f\left(x,y\right)\text{and}g\left(x,y\right)$ are differentiable at the origin and have nonzero gradients there, and that $g\left(0,0\right)=0.$ If the maximum value of $f\left(x,y\right)$ subject to the constraint $g\left(x,y\right)=0$ occurs at the origin, how is the tangent line to the graph of $g\left(x,y\right)=0$ related to the tangent line at the origin to the level curve of f through $\left(0,0\right)?$

(b) Suppose that $f\left(x,y,z\right)\text{and}g\left(x,y,z\right)$ are differentiable at the origin and have nonzero gradients there, and that $g\left(0,0,0\right)=0.$ If the maximum value of $f\left(x,y,z\right)$ subject to the constraint $g\left(x,y,z\right)=0$ occurs at the origin, how is the tangent plane to the graph of the constraint $g\left(x,y,z\right)=0$ related to the tangent plane at the origin to the level surface of f through $\left(0,0,0\right)?$