Does Zero Derivative Always Indicate Maximum or Minimum in Calculus?

[batballbat]

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

 

[HallsofIvy]

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.

 

[batballbat]

but can every local minima be a global minimum? or same for maximum?

 

[batballbat]

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?

 

[HallsofIvy]

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.

 

[HallsofIvy]

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, f(x)= x^3 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.