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True / false ?

In continuous function if we have three stationary points then always it will be case that either one is maxima and two are minima or

one is minima and two are maxima ?

I think it is true but i am concerned about constant fucntion
asked in Calculus by Boss (26.8k points) | 179 views
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Hello Rahul.

here condition is 3 critical point.

your constant function y=c would have infinite critical points and x=c would have 0.
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So for every 3 critical points above is always true?
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yes, it is.
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Can you once check nitish comments below for inflection point?
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Hello rahul , i'm sorry for ignoring that point. yes! he is right.

actually maxima/ minima(local or global) means first derivative=0 but first derivative=0 doesn't mean it's minima maxima point , as it may be inflection which is neither maxima or minima.

like in y=x3 , x=0 is stationary point where first derivative is =0 but that point is neither maxima or minima.

so in that way your claim is wrong.

sorry from my side for wrong answer.

1 Answer

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no, it is not always true..what if graph has a structure like 'W'....it will have one maxima and two minima
answered by Boss (30.6k points)
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But its a form of sin wave W,here every 3 critical points means 2 maxima and 1 minima or vice versa
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@rahul.how sin wave has 2 minima and 1 maxima?can u explain??
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sin wave take maximum and minimum at infinite points...sin is a periodic function...for a periodic function if maxima or minima exist, there are infinite points where it can be taken
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yes nitish.thats what i am saying .and sin has -1 as minima ans +1 as maxima
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@Nitish :- what if graph has a structure like 'W'....it will have one maxima and two minima

If it has one maxima and two minima then it satisfies the statemet?

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sorry, i had seen only latter part of qsn where you have written one minima and two maxima.
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So is this statement always true ? Will it hold for stationary points or only for critical points?My thought that it will be true for every three stationary points and as critical points are also stationary points so holds for them also.Please correct if i missed anything
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It will follow...but one issue, if it is a point of inflexion...where second derivative is zero...inflexion point is stationary because first derivative is zero but it will neither be Maxima nor minima because second derivative is zero

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