Final answer:
To find the critical numbers of the function f(x) = x^(-6) * ln(x), we differentiate it using the product rule and set the derivative equal to zero. Solving the resulting equation, we find that the critical number of the function is e^(1/6).
Step-by-step explanation:
To find the critical numbers of the function f(x) = x^(-6) * ln(x), we must first find the derivative of the function and then determine where the derivative is equal to zero or undefined, since these are the points where the function has a horizontal tangent or is non-differentiable.
The derivative of f(x) can be found using the product rule and the chain rule. The product rule states that if you have a function h(x) = u(x)v(x), the derivative h'(x) is given by u'(x)v(x) + u(x)v'(x). Applying the product rule to our function, let u(x) = x^(-6) and v(x) = ln(x), we get:
u'(x) = -6x^(-7) and v'(x) = 1/x because the derivative of ln(x) with respect to x is 1/x.
Thus, the derivative of f(x) is:
f'(x) = u'(x)v(x) + u(x)v'(x) = (-6x^(-7))(ln(x)) + (x^(-6))(1/x) = -6x^(-7)ln(x) + x^(-7).
To find the critical numbers, we set the derivative equal to zero and solve for x:
-6x^(-7)ln(x) + x^(-7) = 0
x^(-7)(-6ln(x) + 1) = 0
Since x^(-7) ≠ 0 for x > 0, we can divide both sides by x^(-7) and obtain:
-6ln(x) + 1 = 0
ln(x) = 1/6
Applying the property that e^(ln x) = x, we solve for x:
x = e^(1/6)
Therefore, the critical number of the given function is e^(1/6).
Remember that since ln(x) is undefined for x ≤ 0, we only consider x > 0.