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Math Help - differentiable function

  1. #1
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    differentiable function

    If the function f(x) is differentiable

    and

    f(x) =
    ax^3 - 6x ; x <= 1
    bx^2 + 4 ; x > 1

    then a =

    A) 0
    B) 1
    C) -14
    D) -24
    E) 26
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  2. #2
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    Quote Originally Posted by DINOCALC09 View Post
    If the function f(x) is differentiable

    and

    f(x) =
    ax^3 - 6x ; x <= 1
    bx^2 + 4 ; x > 1

    then a =

    A) 0
    B) 1
    C) -14
    D) -24
    E) 26
    As f is differentiable f'(x)=3ax-6 for x<=1 and 2b+4 for x>1.

    But both f must be continuous so at x=1 we must have:

    a-6 = b+4

    and as it is differentiable at x=1 we must also have:

    3a-6 = 2b+4

    Now solve these simultaneous equations for a.

    RonL
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  3. #3
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    is the answer C
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  4. #4
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    Quote Originally Posted by DINOCALC09 View Post
    If the function f(x) is differentiable
    and

    f(x) =
    ax^3 - 6x ; x <= 1
    bx^2 + 4 ; x > 1
    Hello,

    if f is differentiable at x = 1 you have to solve for a and b:

    \begin{array}{lcr}f(1)=g(1)&\text{that means}&a-6=b+4\\f'(1)=g'(1)&\text{that means} &3a-6=2b \end{array}

    which will yield a = -14 and b = -24

    So your answer is correct.
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  5. #5
    GAMMA Mathematics
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    Quote Originally Posted by DINOCALC09 View Post
    is the answer C
    Yes, the answer is C because as CaptainBlack said, to be differentiable, a function must be continuous, and therefore its derivatives should have values and be continuous. This gives you two equations with two unknowns. You probably had the f(x) values equal to each other at one, but you may have not realized that f'(x) should be continuous at that point as well.
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