find critical numbers

**Samantha** · July 26th 2007, 08:16 AM

Find all critical numbers for the function f(x) = $\frac {x-1} {x+3}$

**red_dog** · July 26th 2007, 08:42 AM

The critical points are the roots of $f'(x)$ .
So, you have to calculate $f'(x)$ and solve the equation $f'(x)=0$ .

**Plato** · July 26th 2007, 09:05 AM

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.

**Samantha** · July 30th 2007, 04:05 AM

I didn't get it =(

**galactus** · July 30th 2007, 04:29 AM

Plato gave you a good hint. Find f'(x). Where does f'(x) not exist. What makes the denominator equal to 0?.

**Samantha** · July 30th 2007, 04:40 AM

is it
a) 1
b) 1, -3
c) -3
d) 1, -1
e) None of these

**topsquark** · July 30th 2007, 05:38 AM

Originally Posted by Samantha

$f(x) = \frac {x-1} {x+3}$

Well, what's the derivative of your function?

-Dan

**Samantha** · July 31st 2007, 05:25 AM

Originally Posted by topsquark

Well, what's the derivative of your function?

-Dan

$<br /> \frac {1* (x+3) - 1(x-1)} {(x+3)^2} = \frac {x+3 -x+1} {(x+3)^2} = \frac {4} {(x+3)^2}<br />$

**topsquark** · July 31st 2007, 05:54 AM

Originally Posted by Samantha

$<br /> \frac {1* (x+3) - 1(x-1)} {(x+3)^2} = \frac {x+3 -x+1} {(x+3)^2} = \frac {4} {(x+3)^2}<br />$

Now, where is this 0 or undefined?

-Dan

**curvature** · July 31st 2007, 06:13 AM

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.

**topsquark** · July 31st 2007, 06:16 AM

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

**curvature** · July 31st 2007, 06:35 AM

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.

**Samantha** · July 31st 2007, 07:22 AM

I understand now, thanks =)

**Plato** · July 31st 2007, 07:25 AM

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.

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.

**topsquark** · July 31st 2007, 08:17 AM

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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