y = (9.6x^2)(e^(-0.3x))
Using the product rule, we have:
y' = (19.2x)(e^(-0.3x)) + (9.6x^2)(-0.3e^(-0.3x))
Setting y' to zero, we have:
y' = 0
(19.2x)(e^(-0.3x)) + (9.6x^2)(-0.3e^(-0.3x)) = 0
(2x)(e^(-0.3x)) + (-0.3x^2)(e^(-0.3x)) = 0
(2x - 0.3x^2)(e^(-0.3x)) = 0
(2 - 0.3x)(x)(e^(-0.3x)) = 0
2 - 0.3x = 0 OR x = 0 OR e^(-0.3x)) = 0
0.3x = 2 OR x = 0
x = 2 / 0.3 OR x = 0
x = 20/3 OR x = 0
Note that e^(-0.3x) can never be equal to zero.
Hence, the only critical points are x = 20/3 and x = 0.