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9.  4.21. -4.21.  Find the point on the plane $$z = x − 2y + 3$$  that is closest to the origin, by finding where the square of the distance between $(0,0)$ and a point $(x,y)$ of the plane is at a minimum. Use the matrix of second partial derivatives to show that the point is a local minimum.   
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\end{array} \right]  $$  Because 4 > 0 and (4)(10) − (−4)2 = 24 > 0. So by the Theorem 4.3, it is positive definite. By theorem 4.8, If   $triangledown $\triangledown  f (A) = 0$ and the Hessian matrix [f_{x_i $[f_{x_i  x_j} (A]) (A])$  is positive definite at $A$, then $f (A)$ is a local minimum. Therefore, $f$ has a local minimum at point (-0.5,1) $(-0.5,1)$  10.  -7.32. Let $S$ be the unit sphere centered at the origin in $R^3$. Evaluate the following items, using as little calculation as possible  (a)$\int_{S} 1 d\sigma$  (b)$\int_{S} ||X||^2 d\sigma$  (c) Verify that $\int_{S} x_1^2 d\sigma = \int_{S} x_2^2 d\sigma =\int_{S} x_3^2 d\sigma$ using either a symmetric argument or parametrizations. Can you do this without evaluating them?  (d) Use the result of parts (b) and (c) to deduce the value of $\int_{S} x_1^2 d\sigma$  Answer:  (a)In geometry, $\int_{S} 1 d\sigma$ means the area of the unit sphere in $R ^3$  So $\int_{S} 1 d\sigma= \pi·1^3 =4\pi$