Question

Find the critical numbers of the function.
f(x) = x^-2ln(x)

Answer 1

A critical point is an interior point on the domain of a function f(x) where f'(x)=0 or f'(x) is undefined.

The domain of f(x)=x^(-2*ln(x)) is (0,∞) because ln(x) cannot take non-positive arguments.

Now, we need to find f'(x) and where f'(x)=0 or undefined.

If we take the logarithm on both sides and use implicit differentiation, we can find f'(x).

Let y=x^(-2ln(x))
ln(y)=-2ln(x)*ln(x)
ln(y)=-2(ln(x))²
differentiate with respect to x
(1/y)*(dy/dx)=-2(2ln(x))(1/x)
(dy/dx)=-4y ln(x)/x
f'(x)=-4x^(-2ln(x)) ln(x)/x

We see immediately that f'(x)=0 when x=1 (i.e. ln(x)=0).

Check to see if f'(x) is undefined at any point in the domain of f(x), and any other points where f'(x) = 0.

Check my work.