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Thread: Differentiability: f(x) = cx+d for x<=2, x^2-cx for x>2

  1. #1
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    Differentiability: f(x) = cx+d for x<=2, x^2-cx for x>2

    If
    [tex]f(x)= \left\{ \begin{array}{c}cx+d, \mbox{ } x\leq 2\\ x^2-cx, \mbox{ } x>2[/tex]
    and f'(x) is defined at x=2, what is the value of c+d?

  2. #2
    Senior Member skeeter's Avatar
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    Re: Differentiability

    a function is continuous at x = a if ...

    1. f(a) is defined.
    2. lim{x->a} f(x) exists.
    3. lim{x->a} f(x) = f(a)

    use this definition of continuity for the functions f(x) and f'(x) ... you'll find c and d.

  3. #3
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    Re: Differentiability

    I tried that but then I got the equation
    [tex]2c+d=4-2c[/tex]
    which is where I got stuck.

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    Re: Differentiability

    The derivative should also be equal at x = 2:
    [tex]f&#39;(x) = \left\{ \begin{array}{ll} c & x \leq 2 \\ 2x - c & x > 2[/tex]

    Since f'(x) is defined, the deriative at x = 2 can't simultaneously hold 2 values so you can solve for c and then plug it back into the original equation to solve for d.

  5. #5
    Senior Member skeeter's Avatar
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    Re: Differentiability

    Quote Originally Posted by tjkubo
    I tried that but then I got the equation
    [tex]2c+d=4-2c[/tex]
    which is where I got stuck.
    you did it for f(x) ... did you do it for f'(x)?

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