2. ADDITIONAL PROBLEMS 5 points each: I) Define f: R -> R as follows: f ( x ) : = {0 if x 0 Show that f ' (0) = 0 and f " (0) = 0. You may use without justification the fact that limx--.00 xk /ex = 0...

2. ADDITIONAL PROBLEMS 5 points each: I) Define f: R -> R as follows:
f ( x ) : =
{0 if x < 0="" e="" 1-="" x="" if="" x=""> 0
Show that f ' (0) = 0 and f " (0) = 0. You may use without justification the fact that limx--.00 xk /ex = 0 for all real numbers k.' z) Use the chain rule to show that cost x + sin2 x is constant. You may use the fact that the only functions R -+ R with identically zero derivatives are constant functions.2) Conclude, by choosing a value for x, that cos' x + sin2 x = 1 for all x.
'One can show that f (n) (0) = 0 for all nonnegative integers n. Consequently, f is an example of a function all of whose derivatives equal zero at 0, yet f is not identically zero in any open interval containing 0. Hint: It is not valid to compute f' (x) on (—oo, 0) and (0, oo) and then take the limit as x —¦ 0. This does not take into account the value off at x = 0, which is essential if f is even going to be continuous at x = 0. One must use the definition of the derivative to compute f'(0). Then one can compute