Your derivative is correct.
(b) Critical points are those points on the graph of f(x) where f'(x) is zero. So set the first derivative equal to 0 and solve for x:
- 9 e-9x x3 + 3 e-9x x2 = 0
Since e-9x can never be equal to zero, the only way this equation would be true is if:
- 9 x3 + 3 x2 = 0
x2 (3 - 9x) = 0
x = 0 or x = 1/3
So these are the critical numbers.
(c) We divide the entire number line in three parts based on the two critical points:
-inf < x < 0
0 < x < 1/3
1.3 < x < inf
inf stands for infinity.
Now just pick a number from each of these intervals and plug in to the first derivative and just note the sign of the derivative which says whether the function is decreasing or increasing:
-inf < x < 0 sign of f'(x) = +, thus function is increasing
0 < x < 1/3 sign of f'(x) = +, thus function is increasing
1.3 < x < inf sign of f'(x) = -, thus function is decreasing