increasing/decreasing and convavity

Question

1.) f(x)= x ln x

 

a.) find the intervals on which f is increasing or decreasing

 

b.) find the local max and minimum values of f

 

c.) find the intervals of concavity and the inflection points

Answer 1

a)

 

Set the derivative of f(x) equal to zero.

 

f'(x) = 0

 

lnx + 1 = 0

 

lnx = -1

 

 

Write this as an exponents equation.

 

e^-1 = x

 

1/e = x

 

0.367 = x

 

 

This is a critical value.  Now just evaluate the derivative at x=0 and x=1.  If derivative changes from negative to positive, the f(1/e) is a minimum.   If derivative changes from positive to negative, then f(1/e) is a maximum.  If no sign change, then no local extremas.

 

b)

 

If derivative is positive, then there is increase in the interval.  If derivative is negative, then there is decrease in the interval.

 

 

c)

 

Set second derivative equal to zero.

 

f"(x) = 0

 

(1 / x) = 0

 

Since setting x=0 give undefined value, there is no inflection points.

 

 

 

Note, domain of the function is all positive real numbers including zero.