Question

A flat circular plate has the shape of the region x²+y²≤1. The plate, including the boundary where x²+y²=1, is heated so that the temperature at point (x,y) is given by I(x,y)=x2+2y2−x. a) Find and classify all local min, max and saddle points of I over the given disk

Answer 1

To find local minima, maxima, and saddle points of I(x, y) = x² + 2y² - x over the disk x² + y² ≤ 1, use first and second derivative tests for the interior and Lagrange multipliers for the boundary. Then classify the points based on the second derivative test and the determinant of the Hessian matrix.

Step-by-step explanation:

To find and classify all local minima, maxima, and saddle points of the temperature function I(x, y) = x² + 2y² - x over the disk defined by x² + y² ≤1, we first need to use multivariable calculus techniques, specifically the method of Lagrange multipliers for the boundary and first and second derivative tests for the interior of the disk.

Our strategy involves calculating the gradient of I and setting it equal to zero to find critical points inside the disk. For the boundary, we introduce a Lagrange multiplier λ to consider the constraint g(x, y) = x² + y² - 1 = 0. The resulting equations are solved to find points on the boundary that may correspond to extremes.

The next step is to assess the second partial derivatives of I at the critical points to classify them using the second derivative test. The determinant of the Hessian matrix (comprised of these second partial derivatives) will indicate whether we have a local minimum (positive determinant), local maximum (negative determinant and negative leading derivative), or saddle point (negative determinant).