Fermat''s theorem, Mathematics

Assignment Help:

Fermat's Theorem : If  f ( x ) contain a relative extrema at x = c & f ′ (c ) exists then x = c is a critical point of f ( x ) . Actually, it will be a critical point such that f ′ (c ) = 0 .

Note as well that we can say that f ′ (c ) = 0 since we are also supposing that  f ′ (c ) exists.

This theorem described us that there is a nice relationship between relative extrema and critical points. In fact it will let to get a list of all possible relative extrema.  As a relative extrema have to be a critical point the list of all critical points will give us a list of all possible relative extrema.

Consider the case of f ( x ) = x2 .  We illustrated that this function had a relative minimum at x = 0 in various earlier examples. Hence according to Fermat's theorem x = 0 must be a critical point. The derivative of the function is,

                                                               f ′ ( x ) = 2x

Certain enough x = 0 is a critical point.

Be careful not to use wrongly this theorem.  This doesn't say that a critical point will be a relative extrema.  To illustrate this, consider the following case.

f ( x ) = x3                          f ′ ( x ) = 3x2

Clearly x = 0 is a critical point. Though we know that this function has no relative extrema of any kind.  Thus, critical points do not have to be relative extrema.

Also note as well that this theorem says nothing regarding absolute extrema.  An absolute extrema might or might not be a critical point.


Related Discussions:- Fermat''s theorem

Example of linear equations, Example of Linear Equations: Solve the eq...

Example of Linear Equations: Solve the equation 2x + 9 = 3(x + 4). Solution: Step 1. Using Axiom 2, subtract 3x and 9 from both sides of the equation. 2x + 9 = 3(

Math, 1+3+5+7+9+11+13+15+17+19

1+3+5+7+9+11+13+15+17+19

Infinite, why cant we find the value of 1 upon zero

why cant we find the value of 1 upon zero

Algebra, If a^n+1 + b^n+1/a^n + b^n is the arithmetic mean of a and b then ...

If a^n+1 + b^n+1/a^n + b^n is the arithmetic mean of a and b then find n. Answer:Arithmatic mean of a,b is =(a+b)/2  from the problem (a+b)/2=(a^n+1 +b ^n+1)/(a^n+b^n)  then (a+

Judgment sampling, Judgment Sampling Here the interviewer chooses whom ...

Judgment Sampling Here the interviewer chooses whom to interview believing that their view is more fundamental because they might be directly affected for illustration, to find

DIFFERENTIAL EQUATION, Find an integrating factor for the linear differenti...

Find an integrating factor for the linear differential equation and hence Önd its general solution: SOLVE T^ 2 DY DX+T2

Unitary Method Sample Questions, Where can I find sample questions of Unita...

Where can I find sample questions of Unitary Method for kids to practice? I need  Unitary Method  study material if availbale here on website, i found there is very useful material

Squeeze theorem (sandwich theorem and the pinching theorem), Squeeze Theore...

Squeeze Theorem (Sandwich Theorem and the Pinching Theorem) Assume that for all x on [a, b] (except possibly at x = c ) we have,                                 f ( x )≤ h (

Core concepts, Discuss mareketing core concepts analysing how they are used...

Discuss mareketing core concepts analysing how they are used in marketing hospitality product

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd