A rectangle is inscribed under the curve ( y = 4 - x2 ). Find the dimensions of the rectangle that maximize its area. Use the method of Lagrange multipliers to find the maximum and minimum values of ( f(x, y) = x42 + y\"2 ) subject to the constraint (x + y = 1 ). Find the maximum and minimum values of the function ( f(x) = x*3 - 6x42 + 9x + 1) on the interval ([0, 3]). A box with a square base and open top must have a volume of 32 cubic units. Find the dimensions of the box that minimize the surface area