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Suppose that f is a function whose domain is R and satisfies the following properties:

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Suppose that f is a function whose domain is R and satisfies the following properties: . f(x) = -1 when I < -2 . f(x) = 0 when r > 2 . f(0) = 1. 2 (i) Define the function f on the interval [-2, 2] such that f is everywhere continuous. (ii) Suppose that f must have the form of a quartic polynomial on [-2, 2]; that is, f(x) = car* + c313 + 21" + cir + co. Find the values of co, . . ., c4 such that f is everywhere differentiable. Note: You may choose to give your answers to 5 decimal places instead of writing them as fractions. 2. Consider the equation In ry = (i) Show that (z, y) = (1, 1) is a solution to this equation. (ii) Find dy dr (x,y)=(1,1) (iii) Find dy |(z,y)=(1.1) (iv) Find all point (s) satisfying the equation at which there is a vertical tangent line. 3. Consider the function f(x) = e " on [0, 2], and a point a E [0, 2]. Consider the triangle formed by the tangent line to f at a, and the lines z = 0 and y = 0. Find the point a such that the triangle has largest possible area, and justify that the area has been maximized.