Here's a little trick you can use, if you like: write the denominator in the numerator with opposite sign of the exponent, and use product rule instead. It'll get you the same result, but sometimes it's easier.
f(x) = 8e^3x
f'(x) = 24e^3x
f''(x) = 72e^3x
k(x) = f''(x) / (1 + [f'(x)]^2) ^ (3/2)
k(x) = 72e^3x / (1 + 576e^6x) ^(3/2)
now I know i need to find k'(x) and set it equal to 0 and solve for x, but I just cant seem to come up with an appropriate answer. =/
k'(x) = long complicated quotient rule expression whose value of x escapes me.