Please help me ASAP! Please show all your work and highlight the answer. If there is any problem, please let me know! Thank you :)

Here is my question:

Problem 1: Rectangle of greatest area in an ellipse Use the method of Lagrange multiplier to find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse - + -= 1 with sides parallel to the coordinate axes and symmetric to origin. (Hint: The area of the rectangle is A = (2x) (2y) which needs to be maximized.) Ans: length 2x = 4v2 . height 2y = 3v2 X Problem 2: Ant on Metal Plate The temperature at a point (x, y) on a metal plate is T(x, y) = 4x - 4xy + y?. An ant on the plate walks around the circle x' + y' =52. What are the highest and lowest temperatures encountered by the ant? Ans: 0" at (v5, 2v5) and (-V5, -2v5). 125" at (2v5, -V5) and (-2v5, v5) Video help: https://youtube/LFIRrWu4osl] Double Integral in Cartesian coordinates (P3-5) Problem 3: f(x, y) = =. Integrate f over the region in the first quadrant bounded by lines y = x, y = 2x, x = 1,x = 2. (Ans: =In 2)