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Math Help - find critical numbers

  1. #1
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    find critical numbers

    Find all critical numbers for the function f(x) = \frac {x-1} {x+3}
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    MHF Contributor red_dog's Avatar
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    The critical points are the roots of f'(x).
    So, you have to calculate f'(x) and solve the equation f'(x)=0.
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  3. #3
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    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.
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    I didn't get it =(
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    Eater of Worlds
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    Plato gave you a good hint. Find f'(x). Where does f'(x) not exist. What makes the denominator equal to 0?.
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  6. #6
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    is it
    a) 1
    b) 1, -3
    c) -3
    d) 1, -1
    e) None of these
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Samantha View Post
    f(x) = \frac {x-1} {x+3}
    Well, what's the derivative of your function?

    -Dan
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    Quote Originally Posted by topsquark View Post
    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 />
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  9. #9
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Samantha View Post
     <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
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  10. #10
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    Quote Originally Posted by Plato View Post
    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.
    Attached Thumbnails Attached Thumbnails find critical numbers-july31.gif  
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  11. #11
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by curvature View Post
    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
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  12. #12
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    Quote Originally Posted by topsquark View Post
    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.
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  13. #13
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    I understand now, thanks =)
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  14. #14
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    Quote Originally Posted by Plato View Post
    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 View Post
    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.
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  15. #15
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Plato View Post
    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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