find the critical points of the following function.\n$f(x)=\\frac{1}{x}-\\ln x$\nselect the correct choice…

find the critical points of the following function.\n$f(x)=\\frac{1}{x}-\\ln x$\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the critical point(s) occur(s) at $x=$ (use a comma to separate answers as needed.)\nb. there are no critical points

find the critical points of the following function.\n$f(x)=\\frac{1}{x}-\\ln x$\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the critical point(s) occur(s) at $x=$ (use a comma to separate answers as needed.)\nb. there are no critical points

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x)=\frac{1}{x}-\ln x) (where (x>0) since (\ln x) is defined for (x > 0)) using the power rule ((x^n)^\prime=nx^{n - 1}) and ((\ln x)^\prime=\frac{1}{x}). (f^\prime(x)=-\frac{1}{x^{2}}-\frac{1}{x}=-\frac{1 + x}{x^{2}})

Step2: Set the derivative equal to zero and solve for (x)

Set (f^\prime(x) = 0), so (-\frac{1 + x}{x^{2}}=0). Multiply both sides by (-x^{2}) (since (x^{2}>0) for (x\neq0) and (x>0) in the domain of (f(x))), we get (1 + x=0), which gives (x=- 1). But (x=-1) is not in the domain of (f(x)) ((x>0)). Now, check where the derivative is undefined. The derivative (f^\prime(x)=-\frac{1 + x}{x^{2}}) is undefined when (x = 0), but (x = 0) is not in the domain of (f(x)) (because (\ln x) is undefined at (x = 0)).

Answer:

B. There are no critical points