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Determine whether the following statements are true and give an explanation or counterexample. a. Suppose you are standing at the center of a sphere looking at a po...

Question

Determine whether the following statements are true and give an explanation or counterexample. a. Suppose you are standing at the center of a sphere looking at a point $P$ on the surface of the sphere. Your line of sight to $P$ is orthogonal to the plane tangent to the sphere at $P$. b. At a point that maximizes $f$ on the curve $g(x, y)=0,$ the dot product $\nabla f \cdot \nabla g$ is zero.

Determine whether the following statements are true and give an explanation or counterexample. a. Suppose you are standing at the center of a sphere looking at a point $P$ on the surface of the sphere. Your line of sight to $P$ is orthogonal to the plane tangent to the sphere at $P$. b. At a point that maximizes $f$ on the curve $g(x, y)=0,$ the dot product $\nabla f \cdot \nabla g$ is zero.



Answers

Determine whether the statement is true or false. Explain your answer. If the tangent plane to the level surface of $F(x, y, z)$ at the point $P_{0}\left(x_{0}, y_{0}, z_{0}\right)$ is also tangent to a level surface of $G(x, y, z)$ at $P_{0},$ then $\nabla F\left(x_{0}, y_{0}, z_{0}\right)=\nabla G\left(x_{0}, y_{0}, z_{0}\right)$


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