Here’s a breakdown of the problems in the image: --- ### 1. **Hottest and Coldest Points on a Plate** **Given:** - Plate region: \( x^2 + y^2 \leq 1 \) - Temperature: \( T(x, y) = x^2 + 2y^2 - x \) **Task:** - Find the hottest and coldest points on the plate and the temperature at each. **Approach:** - Find critical points in the interior (solve \( \nabla T = 0 \)). - Check the boundary (\( x^2 + y^2 = 1 \)), possibly using Lagrange multipliers. - Compare values to determine max/min. --- ### 2. **Relative Extremes and Saddle Points** **Given:** - \( f(x, y) = \frac{1}{x} - \frac{27}{y} + xy \) **Task:** - Find all relative extrema and saddle points. - Show all work and explain classification. **Approach:** - Find critical points (\( \nabla f = 0 \)). - Use second derivative test to classify. --- ### 3. **Hottest Point on an Ellipsoid (Lagrange Multipliers)** **Given:** - Ellipsoid: \( 4x^2 + y^2 + 4z^2 = 16 \) - Temperature: \( T(x, y, z) = 8x^2 + 4yz - 16z + 600 \) **Task:** - Find the hottest point on the probe’s surface using Lagrange multipliers. **Approach:** - Maximize \( T(x, y, z) \) subject to the ellipsoid constraint. --- ### **BONUS: Plane in First Octant with Least Volume** **Given:** - Plane passes through \( (3, 2, 1) \) - Find equation of the plane slicing off the least volume in the first octant. **Approach:** - Equation of plane: \( ax + by + cz = d \) passing through (3,2,1). - Minimize the volume below this plane in the first octant. --- Would you like a full solution to any of these problems? If so, which one?

Certainly! Please specify which problem you'd like the full detailed solution for: 1. The hottest and coldest points on the circular plate. 2. Finding all relative extrema and saddle points of \( f(x,y) = \frac{1}{x} - \frac{27}{y} + xy \). 3. The hottest point on the ellipsoid probe using Lagrange multipliers. 4. The equation of the plane passing through (3,2,1) that slices off the least volume in the first octant. Let me know which one you'd like me to work through in detail!