How do I go about addressing this? I understand also just how the Average Value Theorem functions yet I don"t recognize exactly how it have the right to prove that the c is the midsuggest of the interval. Thanks in advanced.
Notice that this especially claims "quadratic function". This isn"t true for just any type of function, f. Write $f(x)= px^2+ qx+ r$. The rest of this is just computation. At x= a that is $pa^2+ qa+ r$. At x= b it is $pb^2+ qb+ r$. $fracf(b)- f(a)b- a= fracpb^2- qb+ r- pa^2- qa+ rb- a= fracp(b^2- a^2)+ q(b- a)b- a= fracp(b+a)(b-a)+ q(b-a)b-a= p(b+a)+ q$.
You are watching: Find the value of c guaranteed by the mean value theorem
On the various other hand $f"= 2px+ q$. The "intend worth theorem" claims that there exit a allude, c, such that $f"(c)= fracf(b)-f(a)b- a$. Here that implies that $2pc+ q= p(b+a)+ q$. From that $2pc= p(b+a)$ so that $c= fracb+a2$
edited Oct 31 "18 at 0:49
answered Oct 31 "18 at 0:39
17.6k22 gold badges1010 silver badges2020 bronze badges
Add a comment |
In order to use MVT, we recognize that $f$ need to be consistent on the closed interval $$ and differentiable on $(a, b)$. Then MVT guarantees that tbelow exists$$f ^ prime ( c ) = frac f ( b ) - f ( a ) b - a $$You desire to present that $c = fraca+b2$ as this is the midallude between $a, b$.
We can verify this with algebraic simplification and also substitution. Let $f(x) = px^2+qx+z$. Then $f"(x) = 2px + q$ such that $a eq 0$.
Plug in $a, b$ in for $x$ in $f(x) = px^2+qx+z$ and then plug that expression into $frac f ( b ) - f ( a ) b - a $.
You have to then get $p(b+a) + q$
Then plug $c$ in for $x$ in $f"(x)$ and you need to gain $pc + q$. Keep in mind that these two worths are equivalent according to MVT. Or if you substitute $fraca+bc$ for $c$ in $f"(x)$ then you will certainly view that the expressions are tantamount.
answered Oct 31 "18 at 0:43
16277 bronze badges
Add a comment |
Thanks for contributing an answer to slrfc.orgematics Stack Exchange!Please be sure to answer the question. Provide details and share your research!
But avoid …Asking for aid, clarification, or responding to other answers.Making statements based on opinion; earlier them up through references or personal experience.
Use slrfc.orgJax to format equations. slrfc.orgJax recommendation.
To learn more, see our tips on writing excellent answers.
See more: Careers For People Who Get Bored Easily (With Salaries), 6 Careers For Those Who Get Bored Easily
Sign up or log in
Sign up utilizing Google
Sign up using Facebook
Sign up using Email and Password
Message as a guest
Email Required, but never shown
Article as a guest
Required, but never shown
Post Your Answer Discard
Not the answer you're looking for? Browse other inquiries tagged calculus or ask your own question.
Featured on Meta
Prove, using the mean worth theorem, that $x+1 lt e^x lt xe^x+1$ for $x gt 0$
Finding all numbers
Proof utilizing the suppose value theorem
Why does the Average Value Theorem call for a closed interval for continuity and an open up interval for differentiability?
Difficulties in stating intend value theorem for features which are not continuous on a closed interval.
For which attributes is the ‘c’ from the expect value theorem the midallude of the interval?
How to usage the Mean-Value Theorem on a function constant over an open interval.
Prove using the Typical Value Theorem
Prove Using L'Hopital's Rule And Typical Value Theorem.
Hot Netoccupational Questions more hot inquiries
Subscribe to RSS
Inquiry feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Stack Exchange Network-related
website architecture / logo © 2021 Stack Exadjust Inc; user contributions licensed under cc by-sa. rev2021.9.16.40224