I\'m getting confused trying to solve this problem: So they want me to find the

Question

I'm getting confused trying to solve this problem:

So they want me to find the critical points (on its domain) and then use the critical points to determine whether it is a local max or local min.

first i solved for the derivative and got: (-x+1) / (x(x-lnx)^2). Then i plugged the graph into the calculator to find its zeros/critical points and found a critical point at x = 1. on the graph of f(x), x = 1 appeared to be a local maximum.

the confusing part is, the answer in the back of the book says that there are no critical points, therefore, no local max/mins. I'm sure that my derivative is right, and the calculator doesnt lie. why is the book saying "no critical points". maybe there is some Critical point law/rule that  don't know about.

f(x)= l/x - lnx So they want me to find the critical points (on its domain) and then use the critical points to determine whether it is a local max or local min. first i solved for the derivative and got: (-x+1) / (x(x-lnx)^2). Then i plugged the graph into the calculator to find its zeros/critical points and found a critical point at x = 1. on the graph of f(x), x = 1 appeared to be a local maximum. the confusing part is, the answer in the back of the book says that there are no critical points, therefore, no local max/mins. I'm sure that my derivative is right, and the calculator doesnt lie. why is the book saying "no critical points". maybe there is some Critical point law/rule that don't know about.

Explanation / Answer

f(x) = 1/x - ln x

f'(x) = -1/x^2 -1/x

For critical point,

-(x+1)/x^2 = 0

x+1=0, x=-1

And for undefined, x=0

So, critical points are -1,0

As domain is given, so, there is no critical points.

Now to determine local maxima minima we need to use second derivative test.

We get f"(x) = x+2)/x^3 And in our domain is it increasing

So no maxima is found

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