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    Position of a Nonnegative Differentiable Function on a Closed Bounded Interval

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    Let f(x), g(x) be functions defined on a closed bounded interval [a, b] such that the following conditions hold:

    g is differentiable on [a, b].

    There are positive constants a, b such that g(x) = a*f(x) - b*(dg/dx).

    f(x) > 0 for all x in [a, b]

    g(x) >= 0 for all x in [a, b]

    g(a) > 0


    Question 1: At how many points of [a, b] could g(x) possibly be equal to 0:

    i. none

    ii. just one

    iii. finitely many but more than one

    iv. countably infinitely many

    v. uncountably many

    Question 2: If the answer to Question 1 is something other than "i," at what point(s) of [a, b] could g(x) possibly be equal to 0?

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    Solution Preview

    Since g is differentiable on the closed bounded interval [a, b], g is continuous on that interval. Thus g attains a minimum somewhere on [a, b].

    If g attains a minimum at x = a, then (since g(a) > 0) g is positive on all of [a, b].

    Now suppose ...

    Solution Summary

    All the characteristics of the given functions f and g, together with general results in calculus, are used to determine whether (and where) the function g could possibly be 0.