I am trying to find the critical values of ln((x^2+4)^(1/2)

Question

I found they are 0,2, and -2. I just wanted some conformation.?  

Answer 1

The function f(x) = ln(√(x² + 4)

Apply derivative with respect to x.

d/dx[f(x)] = d/dx[ln(√(x² + 4)]

Recall the formulae: d/dx[ln(x)] = 1/x

d/dx[√x] = 1/(2√x)

d/dx[xⁿ] = 1/x^n-1

d/dx[constant] = 0

f'(x) = [1/√(x² + 4)] [1/2(√(x² + 4))] [2x]

f'(x) = 2x/[2(x² + 4)]

f'(x) = x/(x² + 4)

f'(x) is undefined at x = 2 and - 2 is included in the domain of f .

To find critical values equate the first derivative to 0.

x/(x² + 4) = 0

x = 0

Critical points are at x = 0, 2 and - 2.