Critical Numbers

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November 20th 2009, 05:56 PM
rawkstar

Critical Numbers

so I tried doing this problem but I'm having difficulty, please help

find all critical numbers for the function f(x)=(9-x^2)^(3/5)

First thing i did was to find the derivative which i got -6x/[5(9-x^2)^(2/5)]

The next thing was to find when f'(x)=0, so i set the derivative equal to 0 to get x=0

I then checked to see when the derivative is undefined. To do this i set the denominator of the derivative [5(9-x^2)^(2/5)] to zero and got x=3 and x=-3.

are x=0,-3,3 right? I graphed the original function on my calculator and can't find anything specific about these points. also, i used the derivative function on my calculator to get the derivatives at these points, i thought that i'm not supposed to be able to do this since the derivative is supposed to be undefined at x=3 and x=-3
November 20th 2009, 06:14 PM
ANDS!

The original function you have is defined over all real numbers. There is no need to check the domain of the derivative.
November 20th 2009, 06:25 PM
rawkstar

how do you know that it is defined over all real numbers, i thought it would be undefined when the denominator equals 0
November 20th 2009, 06:38 PM
Progenitor12

rawkstar,

I think you're right. The derivative is 0 when x=0 and undefined when x={-3,3}.

I graphed the function and it appears to have vertical asymptotes at x={-3,3} and it passes through 0 at x=0.
November 20th 2009, 06:46 PM
rawkstar

the problem with mine is when i graph it, i am not getting the asymptotes at x=3 and x=-3
November 20th 2009, 06:53 PM
Progenitor12

http://img9.imageshack.us/img9/910/graphif.th.jpg
November 20th 2009, 06:58 PM
rawkstar

ok i had a typo with my calculator input
i see it now, thank you
November 20th 2009, 07:03 PM
ANDS!

Quote:

Originally Posted by rawkstar

how do you know that it is defined over all real numbers, i thought it would be undefined when the denominator equals 0

F(X) is a composition of functions. Each of which is defined over the reals. The derivative is just the derivative.
November 20th 2009, 07:11 PM
Progenitor12

Quote:

Originally Posted by ANDS!

F(X) is a composition of functions. Each of which is defined over the reals. The derivative is just the derivative.

The function is undefined when the denominator is 0, that is, when $9-x^{2}=0$ .

Even if two functions are defined for all x, their composed ratio is not *necessarily* defined for all x. It is undefined when the denominator is 0, just like any other fraction.
November 20th 2009, 07:19 PM
ANDS!

Quote:

Originally Posted by Progenitor12

The function is undefined when the denominator is 0, that is, when $9-x^{2}=0$ .

Even if two functions are defined for all x, their composed ratio is not *necessarily* defined for all x. It is undefined when the denominator is 0, just like any other fraction.

The original function is most definitely defined over the reals, regardless of the domain of it's derivative.

The behavior of the derivative (including its domain) will help describe the behavior of the original function, but the two are not necessarily linked as far as domain is concerned.