MATH 156, Winter 2024, Written Assignment 5 Please submit your full solutions before 4:00pm on Thursday, February 15, 2024 through Assign2. 1. (2+3=5 points/ For the function f(x, y) = In(1 + 2 +) (a) Find all critical points. (b) Use the second derivative test to clarify the nature of each point, if possible. 2. [2+3=5 points/ For the function f (x, y) = ("thy(y? -z?) (a) Find all critical points. (b) Determine, if possible, whether f has a relative minimum, relative maximum or saddle point at each critical point. 3. (3 points) Find the gradient ", " ) of f(x, y, 2) = yet and evaluate it at the point (0, 1, -1). 4. /2+3+2=7 points/ In this problem, we will use the multi-variable chain rule to derive the (single variable) product rule. We let S(t) and g(t) be differentiable functions and set h(t) = f(t)g(t). Let P(x, y) = ry. (a) Find the partial derivatives- op and ap dy (b) Apply the multi-variable chain rule to P(x, y), with a = f(t) and y = g(t), to find . Your final answer should be expressed only in terms of the independent dt variable t (this is, it shouldn't contain r and y). (c) Show that h(t) = P(f(t), g(t)), and use part (b) to show