Critical Points of a Function: Finding the Points of Inflection

[Niaboc67]

Homework Statement

1.) Let g(x) = x − (5/x^2)

find all critical numbers (if any) of g.

Give answers in increasing order (smallest first). Enter DNE in any unused space.

x =

x =

and

2.) Let g(x) = x |x + 5|

The Attempt at a Solution

[/b]

I know I must first take the derivative and then set to zero, in order to find it's zero.

g(x) = x - 5/x^2
g'(x) = 1+10x^-3
g'(x) = 10/x^3 = -1
g'(x) = (x^3)10/x^-3 = -1(x^3)
g'(x) = 10 = -1(x^3)
g'(x) = -10 = x^3
g'(x) = -10^(1/3) = x
After here I don't know what I am suppose to do in order to find the crical points :/

 

[andrewkirk]

Critical points are where the derivative is zero or undefined. You've found where it's zero. Now, for what value(s) of x is it undefined?

 

[RJLiberator]

That seems to be right. You have a correct x value from my understanding. To find the point, you input x into the correct function to receive the y output.

edit: Andrew posted a better response.

 

[SammyS]

Homework Statement

1.) Let g(x) = x − (5/x^2)

find all critical numbers (if any) of g.

Give answers in increasing order (smallest first). Enter DNE in any unused space.
x =
x =
and

2.) Let g(x) = x |x + 5|

The Attempt at a Solution

[ B ]
I know I must first take the derivative and then set to zero, in order to find it's zero.

g(x) = x - 5/x^2
g'(x) = 1+10x^-3

That is correct for the derivative, g'(x) .

After this you apparently meant to set g'(x) = 0 and then solve for x. However, you kept setting each line equal to g'(x).

None of the following lines is equal to g'(x).

g'(x) = 10/x^3 = -1
g'(x) = (x^3)10/x^-3 = -1(x^3)
g'(x) = 10 = -1(x^3)
g'(x) = -10 = x^3
g'(x) = -10^(1/3) = x
After here I don't know what I am suppose to do in order to find the critical points :/[ /B ]

There are many contexts for critical numbers.

Here it appears that you are trying to find places at which the derivative can change sign.

Essentially, you need to identify places where the derivative is zero and places where the derivative is undefined.

 

[Ray Vickson]

Critical points are where the derivative is zero or undefined. You've found where it's zero. Now, for what value(s) of x is it undefined?

Does that definition (non-existence of derivative) apply also at points where the function itself is undefined? That is the case here!

 

[Mark44]

None of the following lines is equal to g'(x).

g'(x) = 10/x^3 = -1
g'(x) = (x^3)10/x^-3 = -1(x^3)
g'(x) = 10 = -1(x^3)
g'(x) = -10 = x^3
g'(x) = -10^(1/3) = x
After here I don't know what I am suppose to do in order to find the critical points :/

To elaborate on what SammyS said, the last correct line you (Niaboc67) wrote before the first line quoted above was
g'(x) = 1+10x^-3

The next step should be to set g'(x) = 0
##g'(x) = 0 \Rightarrow 1 + 10x^{-3} = 0##
Now, work with that.

 

[andrewkirk]

Does that definition (non-existence of derivative) apply also at points where the function itself is undefined? That is the case here!

I presume so, as the function being undefined at ##x=a## necessarily implies that the derivative is undefined there, even if ##\lim_{x\to a}f'(x)## exists.

To me it seems more natural to define a critical point as one where the derivative is zero or the function is undefined. A difference would be cases like ##y=x^\frac{1}{3}##, which would be critical at at ##x=0## under the first definition but not under that alternative.

 

[ehild]

The critical point must belong to the domain of the function. Think of the function g(x)=√x. Are all x<0 critical points? x=0 is the only critical point, as the function exists there but its derivative does not exist.

We say that [PLAIN]http://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints_files/eq0001M.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints_files/empty.gif is a critical point of the function f(x) if f(x) exists and if either of the following are true: f '(c) =0 or f '(c) does not exist. [PLAIN]http://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints_files/empty.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints_files/empty.gif