Question #31214

1 Answer
Jan 18, 2017

Answer:

See the explanation section, below.

Explanation:

A Surge Function has the form #f(x) = axe^(-bx)# for positive #a,b#.

In order to find the maximum, we must find the derivative and the critical numbers for #f#.

#f'(x) = ae^(-bx) - abxe^(-bx) = ae^(-bx)(1-bx)#.

#f'(x) = 0# when #1-bx =0#. Which happens at #x=1/b#.

We know that #a# and #e^(-bx)# are both positive, so the sign of #f'(x)# agrees with that of #(1-bx)#.

#f'(x) < 0# for #x < 1/b# (test #1/(2b)#)
and
#f'(x) > 0# for #x > 1/b# (test #2/b#).

Therefore, #f(1/b) = a/(be)# is the maximum.