Define the function
f(x)=⎩⎨⎧x2sin(x1),0,x=0,x=0.
A Maclaurin polynomial of degree 5 will be constructed for f.
(1) Part 1: (a) Show that f is differentiable at x=0 and compute f′(0). Provide full justification, including a limit calculation.
(2) Part 2: (b) Compute f′′(0), f(3)(0), f(4)(0), and f(5)(0), or explain why any of these derivatives does not exist. Support each conclusion with appropriate limits.
(3) Part 3: (c) Using your results from parts (a) and (b), write the fifth-degree Maclaurin polynomial P5(x) for f(x).
(4) Part 4: (d) State whether any non-zero terms of degree higher than 5 can appear in the Maclaurin series for f. Justify your answer based on the behavior of the derivatives at x=0. No computation beyond fifth order is required.