f(x) = x / (x + 7)
f'(x)
= [(x + 7)(1) - (x)(1)] / (x + 7)^2
= [x + 7 - x] / (x + 7)^2
= 7 / (x + 7)^2
When f'(x) = 6, we have:
7 / (x + 7)^2 = 6
(x + 7)^2 = 7 / 6
x + 7 = sqrt(7/6) or = -sqrt(7/6)
x = -7 + sqrt(7/6) or x = -7 - sqrt(7/6)
Hence, the greater value is x = -7 + sqrt(7/6) and the lesser value is x = -7 - sqrt(7/6).