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Math Help - differentiation

  1. #1
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    differentiation

    If f is differentiable and the difference quotient overestimates the slope of f at x=a for all h>0, which must be true?

    a) f '(a)>0
    b) f '(a)<0
    c) f ''(a)>0
    d) f ''(a)<0

    i don't understand the concept of this question. can someone help?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by got_jane View Post
    If f is differentiable and the difference quotient overestimates the slope of f at x=a for all h>0, which must be true?

    a) f '(a)>0
    b) f '(a)<0
    c) f ''(a)>0
    d) f ''(a)<0

    i don't understand the concept of this question. can someone help?
    You are told that:

    \frac{f(a+h)-f(a)}{h}>f'(a), \ \forall h>0.

    But the answer is none of the the offered options, for consider f(x)=x^2, and a=0 , the difference quotient is:

    \frac{h^2}{h}=h>0, \ \forall h>0

    but f'(a)=0, and f''(a)=0.

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    You are told that:

    \frac{f(a+h)-f(a)}{h}>f'(a), \ \forall h>0.

    But the answer is none of the the offered options, for consider f(x)=x^2, and a=0 , the difference quotient is:

    \frac{h^2}{h}=h>0, \ \forall h>0

    but f'(a)=0, and f''(a)=0.

    RonL
    I believe the answer is f"(a) > 0 (meaning f'(a) is an increasing function). In the example you just offered at x = 0, f"(a) = 2 not 0.
    Last edited by h2osprey; April 2nd 2008 at 10:56 PM.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by h2osprey View Post
    I believe the answer is f"(a) > 0 (meaning f'(a) is an increasing function). In the example you just offered at x = 0, f"(a) = 2 not 0.

    Brain fart on my part, try f(x)=x^3, f'(x)=3x^2, f''(x)=6x, so the difference quotient at x=0 is h^3>0 for h>0, but the first two derivatives are zero.

    (the problem arrose because of a last minute replacement of a much more interesting counter example, the point was that you can construct a function satisfying the required condition but which satisfy none of the given candidates for the answer, as an alternative consider f(x)=x^4)

    In fact the conditions on f'' cannot apply even if assorted ">"s are changed to ">="'s because there is no condition requiring f'' to exist.

    Now there may be a formulation of a question like this that does have an answer like c, but this is not that question.

    RonL
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  5. #5
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    thank you so much! the both of you!
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