1. We define a function f : [1,1] → R by sin x if 0 < x < 1, I In (x² + 1) + a√1+x+ßx if−1≤x≤ 0, where a, ß ER are constants such that f is continuous and also differentiable at x = 0. f(x) = (a) Determine f'(x) for every z € (-1,0) U (0, 1). Find the constants a and 3 such that f has the given properties. What is the value of f'(0)? (b) Let x € (-1,0) be arbitrary. Compute f"(x) and f"(r). Hence, show that f"(x) > 0 for all x € (-1,0). Determine lim (-1)+ f"(x) and limo- f"(x).

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