Please answer the following calculus questions and as best as possible explain, step by step, how to work around these problems and what rule is being used, if applicable. Please write legibly and again, explain how you got to your answer very well. Thank you so much and if you have any questions, don't hesitate to ask! :)

1. Let f (as, y) : 83:3 + y3 + 6mg. Find and classify all critical points. 2. The temperature at a point (11:, y) on a planar metal part of a machine is given by T($, y) = 4:132 4:133; + y2. A moving part of the machine moves around a circle which can be thought as a circle centered at the origin with radius 5 on the machine. Determine the points at which the part will have the highest and lowest temperatures during its motion. Hint: The function may have a critical point on the circle. So, do not forget to analyze the case for which A : 0 when you use Lagrange multi pliers. 3. Let f(:1:,y) = $4 + 43:2(y 2) + 8(y 1)2. Find and classify all critical points. 4. Find the shortest distance from the origin to the curve 2:6 + 3,02 = 1