# Paper Homework 6

Problem 2: Many students did the correct calculation on $\nabla f(1,0)=(2,1)$. The next thing to do is to let the direction be $(a,b)$ with $2a+b=1$ and $a^2+b^2=1$. The rest is to solve the equations.

Problem 4: Many students find the correct direction of the normal line $(8,4,8)$. However some of them failed to finish the work. The most efficient way to do in this problem is to express the normal line in its parametric form $x=8t+1, y=4t+2, z=8t+1$ and plug them into the equation of the sphere, solve $t$ and plug back to get the coordinates of the intersections.

Problem 7: The best thing to do is to ‘eliminate the constraint’. As we want to minimize $x^2+y^2+z^2$ subject to the constraint $y^2=9+xz$, we can just substitute $y^2$ by $9+xz$. Then simply optimize $x^2+9+xz+z^2$. Once we get the critical points, we must compare the values at all critical points we find. For completion, we also need to compare those values with the value of a non-critical point.

Problem 8: Once we find the critical point and evaluate, it is crucial that we compare the value with the value of a non-critical point to see it is a maximum. Strictly speaking, doing the comparison of values can not guarantee the critical point is a maximum or a minimum point, but since proving a point is the global extreme over an unbounded region is beyond this class, we only expect you to do this simple check. Alternatively, one can also use the second derivative to check.