Question 1 [15 marks in total] Each part defines a function f on a subset A...

 

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Question 1 [15 marks in total] Each part defines a function f on a subset A of R². Determine whether A is a convex set, and if so, whether f is a concave function, a convex function, or neither. 1. [5 marks] A = R², f(x, y) = log(x² + y² + 1). 2. [5 marks] A = R² \ {0}, f(x, y) = ²² + 3. [5 marks] A = {(x, y, z) = R³ : x² + y² <1 and x + y −z≥ −5}, f(x, y, z) = xyz. Question 1 [15 marks in total] Each part defines a function f on a subset A of R². Determine whether A is a convex set, and if so, whether f is a concave function, a convex function, or neither. 1. [5 marks] A = R², f(x, y) = log(x² + y² + 1). 2. [5 marks] A = R² \ {0}, f(x, y) = ²² + 3. [5 marks] A = {(x, y, z) = R³ : x² + y² <1 and x + y −z≥ −5}, f(x, y, z) = xyz.