Consider the function f : x |→ 1/x * ln x.

Question

(a) Find the lim x→0
(b) Find the interval on which f is increasing.

(c) Find the point at witch f attains its maximum value

Answer 1

Correction:  at WHICH f attains its maximum...   (The kind of witch you used is a broom-rider.)

 

lim (as x approaches 0⁺) of f(x) = (ln x)/x

 

a) The logarithm goes to -infinity and the factor 1/x goes to positive infinity so their product goes to negative infinity.

 

b) f(x) = x^-1 ln x   so   f'(x) = -x^-2ln x + x^-1(1/x) = x^-2[1 - ln x]

 

On the domain (0,infinity) the first factor of f' is always positive, so the critical point is where ln x = 1, so x = e is the C.P.

 

f(x) is positive on (0,e) and negative on (e,infinity) which means the function is increasing, then decreasing when moving from left to right toward & through the critical point. Therefore the maximum function value is when x=e

and this maximum point is (e,1/e).