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Math Help - differentiation problem help/check work

  1. #1
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    differentiation problem help/check work

    find all values of the constants a and b for which the function

    f(x)=ax, x<2 and ax^2-bx+3, x greater than or equal to 2

    will be differentiable for all values of x

    so what i first did was to find the values of a and b for which the function is continuous, so i took the left sided and right sided limits as they approached 2
    for the left sided limit i got 2a, and for the right sided limit i got 4a-2b+3
    i set them equal to each other 2a=4a-2b+3 and rearranged to get -2a+2b=3
    next i found the left sided and right sided limits of the derivative of the function as it approached 2
    for the left side limit i got 2, and for the right side limit i got 4a-b
    setting them equal to each other to get 4a-b=2, i then have a system of equations to solve
    -2a+2b=3
    4a-b=2
    solving i get a=7/6 and b=8/3

    i just want someone to look over my work and verify that its right or wrong, or if this is how someone would go about solving a problem like this, i wasnt really sure, my first guess was that i had to use the limit definition of the derivative to somehow get the answer, but i wasnt really sure if that would work so i tried the method used above.
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  2. #2
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    Yes it's correct.
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  3. #3
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    thanks
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Don't forget to study if f is differentiable at x_0 =/= 2 as the question says.

    For example if x_0 < 2 there exists a neighborhood V of x_0 such that f : V -> IR , f ( x ) = a x so, f is an elemental function on V and by a well known theorem f' ( x ) = a etc.
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