# find critical numbers

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• Jul 26th 2007, 08:16 AM
Samantha
find critical numbers
Find all critical numbers for the function f(x) = $\frac {x-1} {x+3}$
• Jul 26th 2007, 08:42 AM
red_dog
The critical points are the roots of $f'(x)$.
So, you have to calculate $f'(x)$ and solve the equation $f'(x)=0$.
• Jul 26th 2007, 09:05 AM
Plato
Definitions do differ. In North America it is common to see the following:
A critical number of a function f is a number c in the domain of f at which f ’(c)=0 or f ’(c) does not exist.
• Jul 30th 2007, 04:05 AM
Samantha
I didn't get it =(
• Jul 30th 2007, 04:29 AM
galactus
Plato gave you a good hint. Find f'(x). Where does f'(x) not exist. What makes the denominator equal to 0?.
• Jul 30th 2007, 04:40 AM
Samantha
is it
a) 1
b) 1, -3
c) -3
d) 1, -1
e) None of these
• Jul 30th 2007, 05:38 AM
topsquark
Quote:

Originally Posted by Samantha
$f(x) = \frac {x-1} {x+3}$

Well, what's the derivative of your function?

-Dan
• Jul 31st 2007, 05:25 AM
Samantha
Quote:

Originally Posted by topsquark
Well, what's the derivative of your function?

-Dan

$
\frac {1* (x+3) - 1(x-1)} {(x+3)^2} = \frac {x+3 -x+1} {(x+3)^2} = \frac {4} {(x+3)^2}
$
• Jul 31st 2007, 05:54 AM
topsquark
Quote:

Originally Posted by Samantha
$
\frac {1* (x+3) - 1(x-1)} {(x+3)^2} = \frac {x+3 -x+1} {(x+3)^2} = \frac {4} {(x+3)^2}
$

Now, where is this 0 or undefined?

-Dan
• Jul 31st 2007, 06:13 AM
curvature
Quote:

Originally Posted by Plato
Definitions do differ. In North America it is common to see the following:
A critical number of a function f is a number c in the domain of f at which f ’(c)=0 or f ’(c) does not exist.

Since x=-3 is not in the domain of the function and there no c such that f'(c)=0, this function has no critical numbers.
• Jul 31st 2007, 06:16 AM
topsquark
Quote:

Originally Posted by curvature
Since x=-3 is not in the domain of the function and there no c such that f'(c)=0, this function has no critical numbers.

But by definition I believe x = -3 still gets to be a critical number. (As Plato says, it depends on how we define critical numbers.) I agree that this is probably a philosophical point in this case.

-Dan
• Jul 31st 2007, 06:35 AM
curvature
Quote:

Originally Posted by topsquark
But by definition I believe x = -3 still gets to be a critical number.

I agree with you if we think a critical number is a point where the monotonicity of a function might change.
• Jul 31st 2007, 07:22 AM
Samantha
I understand now, thanks =)
• Jul 31st 2007, 07:25 AM
Plato
Quote:

Originally Posted by Plato
A critical number of a function f is a number c in the domain of f at which f ’(c)=0 or f ’(c) does not exist.

Quote:

Originally Posted by topsquark
But by definition I believe x = -3 still gets to be a critical number… agree that this is probably a philosophical point in this case. Dan

I agree with Curvature that the function has no critical numbers. I do not think that it is a philosophical point.

Here is list of the most widely used calculus texts. I have listed them in order of popularity: Stewart; Larson, Hostetler & Edwards; Hughes-Hallett, Gleason, McCallun; Thomas/Finney; Varberg & Purcell; Smith & Minton; and (out of print) Sallas & Hille. In all of these the definition of critical number is exactly as I gave it above. The one exception is found in Thomas/Finney and they require c to be an interior point of the domain. It seems to me this makes a convincing argument that –3 is not a critical number for this function.
• Jul 31st 2007, 08:17 AM
topsquark
Quote:

Originally Posted by Plato
I agree with Curvature that the function has no critical numbers. I do not think that it is a philosophical point.

Here is list of the most widely used calculus texts. I have listed them in order of popularity: Stewart; Larson, Hostetler & Edwards; Hughes-Hallett, Gleason, McCallun; Thomas/Finney; Varberg & Purcell; Smith & Minton; and (out of print) Sallas & Hille. In all of these the definition of critical number is exactly as I gave it above. The one exception is found in Thomas/Finney and they require c to be an interior point of the domain. It seems to me this makes a convincing argument that –3 is not a critical number for this function.

Fair enough. I accept the correction. :)

-Dan
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