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Thread: Intervals

  1. #1
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    Talking Intervals

    need help with few questions:

    1) Find all open intervals on which $\displaystyle f(x)=\frac{x^2}{x^2+4}$ is decreasing.
    this is what i did:
    a. fount the derivative: $\displaystyle f'(x)=\frac{8x}{(x^2+4)^2}$
    b. set it equal to zero: $\displaystyle 8x=0$
    c. $\displaystyle -{\infty}<x<0$ decreasing
    is that all you do? I'm concern for this problem b/c they say to "Find all open intervals".

    2) Consider $\displaystyle f(x)=x^{1/2}$. Find all value(s), c, in the interval [0,1] such that the slope of the tangent line to the graph of f at c is parallel to the secant line through the point (0, f(0)) and (1, f(1)).
    $\displaystyle \frac{1}{2x^{1/2}}$ i have no idea what to do next...

    3) Explain why $\displaystyle f(x)=\frac{1}{x}$ has a minimum on the interval [1,2] but not on the interval [-1,1].
    graph:

    is it b/c the line is decreasing at interval [1,2]?


    thx in advance
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  2. #2
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    Quote Originally Posted by BryanK View Post
    need help with few questions:

    1) Find all open intervals on which $\displaystyle f(x)=\frac{x^2}{x^2+4}$ is decreasing.
    this is what i did:
    a. fount the derivative: $\displaystyle f'(x)=\frac{8x}{(x^2+4)^2}$
    b. set it equal to zero: $\displaystyle 8x=0$
    c. $\displaystyle -{\infty}<x<0$ decreasing
    is that all you do? I'm concern for this problem b/c they say to "Find all open intervals".
    To find where it is increasing you need to look at the sign of the derivative. Since $\displaystyle f'(x) = 8x/(x^2+4)^2$ and the denominator is positive the sign only depends on the numerator $\displaystyle 8x$. Now if $\displaystyle x>0$ then the numerator is positive and so $\displaystyle f'(x)>0$ which means $\displaystyle f(x)$ is increasing. Similarly, $\displaystyle f(x)<0$ for $\displaystyle x<0$. So it is increasing on $\displaystyle (0,\infty)$ and decreasing $\displaystyle (-\infty,0)$.
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  3. #3
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    Mar 2007
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    um.. u sure that derivative is right?

    i may be horribly mistaken but shouldn’t be:

    x{squared}(x{squared}+4){-1}
    = -x(x{squared}+4){-2}
    :confused:
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