How to find local maxima and minimas of undefined functions?

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SUMMARY

The discussion focuses on finding local maxima and minima for the functions g(x) = |f(x)| and h(x) = +√f(x) over the interval −∞ < x < ∞. It is established that g(x) is always non-negative, implying that any potential global minima will also be non-negative. However, to determine specific local extrema, additional information about the function f(x) is necessary, particularly regarding its continuity and differentiability. Without such details, no definitive conclusions can be drawn about the local extrema of g(x) and h(x).

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  • Understanding of calculus concepts, specifically local maxima and minima.
  • Familiarity with absolute value functions and square root functions.
  • Knowledge of continuity and differentiability of functions.
  • Basic understanding of function properties over intervals.
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  • Research the properties of continuous functions and their implications on local extrema.
  • Study the application of the first and second derivative tests for finding local maxima and minima.
  • Learn about the implications of non-negative functions in optimization problems.
  • Explore examples of piecewise functions to understand how to analyze local extrema in different scenarios.
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Students studying calculus, mathematicians interested in optimization problems, and educators teaching function analysis and extrema.

jackgartlqn
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Hi,
I have been going over past papers and i found this question. Find any local maxima or local minima of the functions g(x) = |f(x)| and h(x) = +√f(x) in the interval, −∞ < x < ∞.
How Would I go about solving this?
 
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jackgartlqn said:
Hi,
I have been going over past papers and i found this question. Find any local maxima or local minima of the functions g(x) = |f(x)| and h(x) = +√f(x) in the interval, −∞ < x < ∞.
How Would I go about solving this?

Hey jackgartlqn and welcome to the forums.

You're going to need information about f(x). You will need some kind of information to start to answer this question in any kind of detail.

We do know that g(x) >= 0 no matter what which means in terms of a potential global minima it is always going to be >= 0 and if f(x) is >= 0 then we can say the same for h(x) as well.


Other than this though, you will need some kind of specific information for f(x) or at least for the properties of f(x).
 
jackgartlqn said:
Hi,
I have been going over past papers and i found this question. Find any local maxima or local minima of the functions g(x) = |f(x)| and h(x) = +√f(x) in the interval, −∞ < x < ∞.
How Would I go about solving this?



But for the fact that it MUST be f(x)\geq 0\,\,\forall x\in(-\infty,\infty) if h(x) is well defined, you can't deduce anything at all if you're not given

some more information about f(x) ...continuity, derivability,...?

DonAntonio
 

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