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Thread: derivatives

  1. #1
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    derivatives

    f(x)=ax^5 when x<f
    f(x)=bx-4 when x>=f

    for f(x) to be differentiable when x=f what should a and b be?

    tysm
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  2. #2
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    Re: derivatives

    f(x)=ax^5 when x<f
    f(x)=bx-4 when x>=f
    let's call the x-value of interest $c$ instead of $f$, so as not to confuse $f$ and $y = f(x)$ ...

    first, the function, $f(x)$, has to be continuous at $\displaystyle x=c \implies \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \implies ac^5 = bc-4$

    second, the function, $f'(x)$, must also be continuous at $\displaystyle x=c \implies \lim_{x \to c^-} f'(x) = \lim_{x \to c^+} f'(x) \implies 5ac^4 = b$

    can you finish?
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    Re: derivatives

    Quote Originally Posted by skeeter View Post
    let's call the x-value of interest $c$ instead of $f$, so as not to confuse $f$ and $y = f(x)$ ...

    first, the function, $f(x)$, has to be continuous at $\displaystyle x=c \implies \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \implies ac^5 = bc-4$

    second, the function, $f'(x)$, must also be continuous at $\displaystyle x=c \implies \lim_{x \to c^-} f'(x) = \lim_{x \to c^+} f'(x) \implies 5ac^4 = b$

    can you finish?
    should the answer be in terms of f ?
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  4. #4
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    Re: derivatives

    Quote Originally Posted by DiscreteMathHelp View Post
    should the answer be in terms of f ?
    yes ... change $c$ to $f$ when you are done.
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