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Math Help - Does this necessarily follow?

  1. #1
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    Does this necessarily follow?

    Assume that f and g are continuous on [a,b] and

    \int_a^b f(x) dx > \int_a^b g(x) dx

    Does it necessarily follow that f(x)>g(x) for all x\in[a,b]?
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  2. #2
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    No
    Take for instance f(x) = 1/2 and g(x) = x over [-1,1]
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  3. #3
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    No, put f(x)=\frac1{1+x^2} and g(x)=x for 0\le x\le1.

    But \frac1{1+x^2}\not>x. (At least not always.)

    (Anyway, the converse is true.)
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  4. #4
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    If\ f(x)>g(x)\ for\ all\ x\ over\ an\ interval,

    then\ \int{f(x)dx}>\int{g(x)dx}\ certainly,\ over\ the\ interval.

    The converse is not true however.
    Similar to the graph given by running-gag,

    f(x)=4x\ from\ x=0\ to\ x=1

    when integrated, gives 2.

    g(x)=x+1

    when integrated over the same interval gives \frac{3}{2}

    f(x)<g(x)\ from\ x=0\ to\ x=\frac{1}{3}

    but

    f(x)>g(x)\ from\ x=\frac{1}{3}\ to\ x=1.
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