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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

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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.

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