Part (a): Finding Critical Points and Their Classification Step 1: Compute Partial Derivatives The function given is ( T(x, y) = \frac{1}{3}x^3 \frac{5}{2}y^2 - 5xy 4x 50 ). First, we need to find the first partial derivatives ( T_x ) and ( T_y ): [ T_x = \frac{\partial T}{\partial x} = x^2 - 5y 4 ] [ T_y = \frac{\partial T}{\partial y} = 5y - 5x ] Step 2: Set Partial Derivatives to Zero To find the critical points, solve ( T_x = 0 ) and ( T_y = 0 ): [ x^2 - 5y 4 = 0 ] [ 5y - 5x = 0 ] From ( 5y - 5x = 0 ), we get ( y = x ). Substitute ( y = x ) into the first equation: [ x^2 - 5x 4 = 0 ] Solving this quadratic equation, we find: [ x = 1 \quad \text{or} \quad x = 4 ] Thus, ( y = 1 ) and ( y = 4 ) respectively. Step 3: Classify the Critical Points To classify these points, we use the second derivatives and the Hessian matrix ( H ): [ T_{xx} = 2x, \quad T_{yy} = 5, \quad