3. Consider the function f : R2 R, given by f(x1, x2) = ln x1 +...

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3. Consider the function f : R2 R, given by f(x1, x2) = ln x1 + tx2 x1+tx2-1 if x 1; otherwise. where t is a free parameter (a number). = (a) Sketch the level graph of the function f for t = -1, t = 0, and t 1. In particular, for each of the three values of the parameter t, draw the level curves for the values f(x1, x2) of the function equal to -1, 0, and 1. [5 marks] (b) Determine if the function f is continuous. Precisely motivate your answer. [4 marks] (c) For every value of the parameter t > 0, evaluate the gradient of the function f at every point (x1, x2). Determine if the function is continuously differentiable over its domain. Precisely motivate your answer. [4 marks] (d) Prove that the function f is concave, for any value of the parameter t > 0. [7 marks] 3. Consider the function f : R2 R, given by f(x1, x2) = ln x1 + tx2 x1+tx2-1 if x 1; otherwise. where t is a free parameter (a number). = (a) Sketch the level graph of the function f for t = -1, t = 0, and t 1. In particular, for each of the three values of the parameter t, draw the level curves for the values f(x1, x2) of the function equal to -1, 0, and 1. [5 marks] (b) Determine if the function f is continuous. Precisely motivate your answer. [4 marks] (c) For every value of the parameter t > 0, evaluate the gradient of the function f at every point (x1, x2). Determine if the function is continuously differentiable over its domain. Precisely motivate your answer. [4 marks] (d) Prove that the function f is concave, for any value of the parameter t > 0. [7 marks]