Local Maxima Minima

Maxima Minima

Definition (local maxima).

LEt be a function then is said to be a local maximum for if there exist such that

Definition (Local minima).

Let be a function then is said to be a local minimum for if there exist such that

Definition ( Local Extrimum ).

is said to be a local exrtimum if it is either local maxima or minima

Definition ( Global Maximum).

Let be a function then is said to be a global maximum if

Definition ( Global minima).

Let be a function then is said to be a global minima if

Definition (Global Extrimum).

If is said to be global extrimum if its either global maximum or minimum

Definition (Interior point).

Let be an interval then is said to be interior point if there exist such that

Theorem (Interior extremum).

Let be a function suppose is an interor point of I and is a local extremum. If is differentiable at then

Proof.

Assume that is a is a local extremum and is differentiable at c
assume there exist such that

as is local maximum for there exist st.

take check then

consider any sequence ( , )
converging to then

hence
then

similarly consider any sequence ( , )
converging to then

hence
then
since

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Theorem (Rolle's).

Let with let be continuous function suppose and is differentiable in Then there exist such that

Proof.

case i) Suppose there exist (a , ) such that then is at some interior point say and from previous theorem

case ii) suppose and for some then hence ther exit some st. note

case iii) =0 for all
then for all
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Theorem ( Mean Value theorem ).

Let with let be a continuous function, diffrentiable in then there exist such that

Proof.

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