Consider the following function f : IR - R defined in terms of two continuous functions a : [0, co) - R and b : (-0o, 0) - R: a(x) for x 2 0 f( z) = b(I) for r Equation A . A- I BIUSX x Styles Font Words: 0 (b) Is it possible to choose f in such a manner that the function F is invertible on R? Answer: Click for List (c) If the function a may be extended to a twice differentiable function on (-E, co) for some e > 0, determine the constants by , b1 , by in b(z) = box2 + biz + bo in terms of a(0), a'(0) and a"(0) such that f is differentiable at z = 0. Syntax advice: Use the notation a0 = a(0), al = a'(0), a2 = a"(0) . Answer: by = (d) In light of the 1% Fundamental Theorem of Calculus, briefly explain in the essay box below why the function F is differentiable at all x * 0 even though, in general, the function f may not be continuous. A . A-I BIUSXX' Styles Font . . .(e) One may show that F is also differentiable at r = 0 with F (0) = 0. Find the derivative F (x) for x 0 in terms of the functions a and/or b. Answer: F(x) = Is the derivative of F continuous at = = 0? Answer: Click for List (f) Find the unique function a such that F (x) = 39xf(20x2) + 69r sin(x2). Answer: a(x) = LT