exponential equation

Question

the equation 3x⁴e^-6x-3x⁵e^-6x =0 has 2 roots 

the smaller root is

the bigger root is 

Answer 1

Rewrite as

 

f(x) = 3*e^-6x(x⁴ -x⁵) = 0

 

In order for this to be true, either 3*e^-6x = 0 or (x⁴ - x⁵) = 0

 

(x⁴ - x⁵) = 0 when x = 1

 

When x = 0, however, the quantity 3x⁴e-^6x - 3x5e^-6x is clearly 0 (by virtue of the x⁴ and x⁵ terms) so the origin (0, 0) is one of the two roots.

 

Note that the quantity 3*e-^6x never truly reaches zero.  It approaches 0 asymptotically at -∞, but fails to actually achieve 0.

 

The smaller root is x = 0

The bigger root is x = 1