a) If b is a positive constant and x > 0, find the critical point of

Question

a) If b is a positive constant and x > 0, find the critical point of f(x) = x − bln(x)

so this part I found to be b.

 

b) using the second derivative test determine if this value is a local maximum, minimum or neither.

I am not sure how to determine this because I do not know how to pick test values without knowing what b is...

Answer 1

Given f(x) = x - b ln(x)

You have found that  f'(x) = 1 - b/x  which gives a critical point of  x  = b.

 

Using the second derivative test, one may be able to determine whether that critical point is a local minimum or a  local maximum.

f"(x) = b/x² 

At x = b,  f"(b) = 1/b > 0 since (b is a positive constant ) b>0.

By the second derivative test, at x = b, there is a local minimum. 

Answer 2

f(x) = x - bln(x)

 

 

Set the derivative equal to zero.

 

1 - b/x = 0

 

x - b = 0

 

x = b

 

 

Second part)

 

Evaluate f'(b - 1) and f'(b + 1).  (b - 1) is the value before x=b.  (b + 1) is the value after x=b.

 

If f'(b - 1) is negative and f'(b + 1) is positive, then f(b) is a minimum.

If f'(b - 1) is positive and f'(b + 1) is negative, then f(b) is a maximum.