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Homework Help: For what value of the constant c is f(x) continuous?

  1. Jul 28, 2008 #1
    1. The problem statement, all variables and given/known data
    For what value of the constant c is the function f continuous on [tex](-\infty,\infty)[/tex]

    [tex]
    f(x)=\left\{\begin{array}{cc}cx^2+2x,&\mbox{ if }
    x<2\\x^3-cx, & \mbox{ if } x\geq2\end{array}\right.
    [/tex]

    2. Relevant equations
    No idea :(


    3. The attempt at a solution
    I tried looking at the examples preceding this section's problem section but could not find anything quite resembling this. There are examples for finding where f would be continous at whatever values at x.. but not for a constant c, which is different. Pointers would be appreciated.
     
  2. jcsd
  3. Jul 28, 2008 #2

    arildno

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    Well, if a function is continuous at a point (here, evidently, the only point of problem is x=2!), then both its one-sided limits must equal the function value AT x=0.

    For a given c, the function value f(2) is given by the lower expression:
    [tex]f(2)=2^{3}-c*2=8-2c[/tex]

    Now, within its domain, the upper expression is just a polynomial in x, i.e continuous.

    That means that f(2) must equal whatever value the upper expression gains AT 2.
    This gives you the equation for c:
    [tex]c*2^{2}+2*2=8-2c[/tex]
    Solve this for c!
     
  4. Jul 28, 2008 #3

    HallsofIvy

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    In order to be continuous, the limit must exist.
    If it does then
    [tex]\lim_{x\rightarrow 2}f(x)= \lim_{x\rightarrow 2^-} f(x)= \lim_{x\rightarrow 2^+}f(x)[/tex]

    Now, what is
    [tex]\lim_{x\rightarrow 2^-} f(x)= \lim_{x\rightarrow 2} cx^2+ 2x[/tex]
    what is
    [tex]\lim_{x\rightarrow 2^+} f(x)= \lim_{x\rightarrow 2} x^3- cx[/itex]

    Set them equal and solve for c.
     
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