Which of the functions below have a local maximum for the function in the x = (0, +∞) domain?

Question

Functions are 1) x!/e^x 2) e^x/x! 3) x^1000000/e^x 4) x^1000000/1000000! 5) 1000000!/x!

Possible multiple choices are:

A) 1, 2, and 3 B) 2, 4, and 5 C) 1 only D) 2 only E) 2 and 3

Answer 1

1.     1) x!/e^x, each integer increase in x increases the numerator by a factor of x but only increases the denominator by a factor of e so the function blows up X => so we are left with B), D), E).

2.     4) x^1000000/1000000! is everywhere increasing on the interval and increases without bound on (0, inf); the denominator is a fixed number => so we are left with D), E).

3.     3) x^1000000/e^x; this function increases from 0 to 1/e and then continues to increase without bound => so we are left with D).

4.     Sure enough, e^x/x! has a maximum since at 0 its value is 1 and at 2, its value is e^2/2 > e^3/3!

5.     I have assumed that the functions with factorials are only defined on integers in the interval.

Answer 2

These kinds of questions are designed to weed out students (or test takers) who rely on just their calculators!

No calculator wants to deal with 1000000! (!).

Strategy: Check the numerator and denominator portions for their behavior (individual slope) as x goes from 0 to +infinity. Comparison shows if a local maximum exists.

Tactics: x! keeps increasing its multiplication factor (x) as x increments. e^x holds its multiplication factor (e) constant as x increments. Since the factorial function is only defined for integral arguments (unlike the gamma function, its close relation), only low values of x need to be checked. (1) will have (1,1/e) and (2,1/e^2) as successive values, the former can't be a local maximum because it's at the extremum of the given domain , for the function. So there will be a local minimum only (at (2, 1/e^2) [above that, x! takes over]. (2) has the flip situation, so it will have a local maximum at (e^2, 2). (3) the x^1000000 term dominates the e^x term utterly, no local mimimum or maximum across the domain (monotonically increases). (4) same as (3); division by the constant 1000000! term doesn't affect the shape of the function graph. (5) the 1/x! is monotonically decreasing across the domain. Again, the 1000000! factor has no effect on the curve shape. So (D) is the answer: only e^x/x! has a local maximum on the interval x = (0, +∞).