Question

Question #31214

Answer 1

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.