Critical Points?

Question

Find the critical points of f(x) = x - 3ln(x)

Answer 1

Or more precisely, the critical point is at:

(3 , 3*(1-ln(3)))

The x coordinate (what is sometimes meant by the critical point) is 3.

Answer 2

3

Answer 3

f'(x) = 1 - 3/x by differentiation rules Note derivative of ln(x) is 1/x
0 = 1 - 3/x Solving for 0 gives critical points
3/x = 1
x = 3 as required.

Plugging into f(x) gives the y coordinate:
f(3) = 3 - 3ln(3) = about -0.296

So the critical point is (3, -0.296)

Answer 4

We have f'(x) = 1 - 3/x , for x >0. So, we have f'(x*)= 0 at x* = 3. Also, f''(x) = 3/x^2, so that f'' is positive all over (0, oo). it's a convex function. This implies f has a global minimum at x* =3 and has no inflections.

Its critical point, a global minimum is so at (3, 3(1 - ln(3))