Does Zero Derivative Always Indicate Maximum or Minimum in Calculus?

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Discussion Overview

The discussion revolves around the conditions under which a zero derivative indicates a maximum or minimum in calculus, particularly in the context of word problems involving optimization of volumes and areas. Participants explore the implications of local versus global extrema and the behavior of linear functions.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether a zero derivative necessarily indicates a local maximum or minimum, suggesting that the nature of the function should also be considered.
  • One participant cites a theorem stating that local maxima or minima occur at points where the derivative is zero, does not exist, or at boundary points, emphasizing that a zero derivative is only one of several conditions.
  • There is a query about whether every local minimum can also be a global minimum, with a response indicating that for this to be true, all local minima must yield the same function value.
  • Another participant raises the case of linear functions, noting that if the derivative is zero over an interval, it does not guarantee a maximum or minimum, as linear functions with non-zero derivatives are either increasing or decreasing.
  • One participant provides an example of a function (f(x) = x^3) where the derivative is zero at a point (x=0) but does not correspond to a maximum or minimum, reinforcing the idea that a zero derivative does not imply an extremum.

Areas of Agreement / Disagreement

Participants express differing views on the implications of a zero derivative, with no consensus reached on whether it always indicates a maximum or minimum. The discussion remains unresolved regarding the conditions under which local extrema can be global extrema.

Contextual Notes

Limitations include the dependence on the definitions of local and global extrema, and the behavior of functions in specific intervals, as well as the potential for misunderstanding the implications of the derivative's value.

batballbat
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in word problems in calculus is see that for finding largest or smallest volumes, areas etc. The derivative is taken zero and the root is found. My question is shouldn't we see the nature of the function? Like even though the function has local minima or maxima, the minimum or maximum can be somewhere else. Or it might not have a zero derivative at all
 
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One of the basic theorems you should have learned is "any local max or min of a function must occur at a point satifying one of three criteria:
1) the derivative is 0 at that point
2) the derivative does not exist at that point
3) the point is a boundary point of the region in question.

The derivative being equal to 0 is only one of those conditions.
 
but can every local minima be a global minimum? or same for maximum?
 
what if it is a linear function? I don't know much calculus, but can anyone tell me the theorem on minima maxima? Is it true that if derivative in an interval is zero then it is maximum or minimum? Or that if it is a maximum or minimum than the derivative is zero?
 
batballbat said:
but can every local minima be a global minimum? or same for maximum?
"Can be", yes. For every local minumum to be a global minimum, they would have to all give the same value of the function, of course.
 
batballbat said:
what if it is a linear function? I don't know much calculus, but can anyone tell me the theorem on minima maxima?
That was what I gave in my first response. A linear function has constant derivative. If that constant is 0, it is a constant function. Every point gives the same value. If that constant is not 0, it is either increasing or decreasing. Since the derivative always exist but is never 0, the max and min must occur at the endpoints of the interval. If there are no endpoints, if the interval is open or infinite, there may be no maximum or minimum.

Is it true that if derivative in an interval is zero then it is maximum or minimum?
No. The fact that a derivative is 0 does not mean there must be a maximum or minimum. For example, [itex]f(x)= x^3[/itex] has derivative 0 at x= 0 but there is no maximum or minimum there.


Or that if it is a maximum or minimum than the derivative is zero?
No. I answered that question in my first response.
 

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