Please answer each step thoroughly

Suppose that f is a function whose domain is R and satisfies the following properties: . f(x) = -1 when I 2 . f(0) = 1. (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 +car + car- + cir + co. Find the values of co, .... ca such that f is everywhere differentiable. Note: You may choose to give your answers to 5 decimal places instead of writing them as fractions. Consider the equation In ry = (i) Show that (z, y) = (1. 1) is a solution to this equation. (ii) Find dr (x, )= (1,1) (iii) Find dr dy (r,9)=(1,1) (iv) Find all point (s) satisfying the equation at which there is a vertical tangent line. 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 a = 0 and y = 0. Find the point a such that the triangle has largest possible area, and justify that the area has been maximized