Find Critical Numbers of g(x) and f(x) Functions

[Skomatth]

I'm supposed to find all critical numbers of the given function.
Book defines critical number c as the place where f'(c)=0 or where f is not differentiable.

1. g(x)= x + 1/x
2. f(x)= x ln (x)

work:
1. g'(x) = \frac{x^2 - 1}{x^2}

c= 1, -1, 0

Zero is wrong however. I put it in the answer because the function is not differentiable at that point. So I'm guessing I have the definition of differntiable at a point wrong. The limit doesn't exist as x approaches 0 so I thought the function wouldn't be differentiable there.

2. f'(x) = lnx + x/x

c= 1/e and all numbers less than or equal to zero.

The book only lists 1/e so same problem here.

 

[MathStudent]

to add to your definition of a critical number, c must also be in the domain of the original function, thus for g(x), 0 is not in the domain of the original function and so it is not a critical number.

 

[dextercioby]

I'm supposed to find all critical numbers of the given function.
Book defines critical number c as the place where f'(c)=0 or where f is not differentiable.

1. g(x)= x + 1/x
2. f(x)= x ln (x)

work:
1. g'(x) = \frac{x^2 - 1}{x^2}

c= 1, -1, 0

Zero is wrong however. I put it in the answer because the function is not differentiable at that point. So I'm guessing I have the definition of differntiable at a point wrong. The limit doesn't exist as x approaches 0 so I thought the function wouldn't be differentiable there.

2. f'(x) = lnx + x/x

c= 1/e and all numbers less than or equal to zero.

The book only lists 1/e so same problem here.

Why...?Are those numbers in the domain of the function "f"...?If so,is the derivative zero...?

Daniel.

 

[Skomatth]

Thx, math student I understand now. I was used to my pre-cal teacher teacher telling me to find critical points to solve rational inequalties which included numbers not in the domain.

 

[dextercioby]

I'm glad you figured out this is something totally different and that the domain of the function is essential.

Daniel.