dfdx=9−1x2. Equating it to 0 , x=±13. f'(x) does not exist at x=0 hence the critical points are −13,0.and+13. For using first derivative test, test the increasing/decreasing behaviour in intervals (−∞,−13),and(13,∞). If f '(x) is positive, then f(x) is increasing and if it is negative, then f(x) is decreasing. Take up any test value say -1 in (−∞,−13),-1/6 in (−13,0),1/6 in (0, 1/3) and +1 in (13,∞)
f ' (x) is positive in (−∞,−13),
negative in (−13,0),
negative in (0,13)and
positive in (13,∞)
Conclusion is there is a relative maxima at x=−13 ( function changes from increasing to decreasing) and relative minima at x= 13 (function changes from decreasing to increasing)
