GATE CSE
First time here? Checkout the FAQ!
x
0 votes
219 views

Question:

The value of $\zeta$ of  $f(b) - f(a) = (b-a) \cdot f'(\zeta)$ for the function $f(x) = Ax^2 + Bx +C$ in the interval $[a,b]$ is _______

 

My Attempt :

STEP 1: $f(x)$ is polynomial function therefore continuous

STEP 2: $f(x)$ is polynomial function therefore differentiable 

STEP 3: There exists a  $c$ such that  $a \leq c \leq b$, such that

$$\begin{align}
f'(c) &= \frac{f(b) - f(a)} {b - a} \\[1em]
f'(c) &= 2Ax + B \\[1em]
\hline
f(b) - f(a) &= (b - a) \cdot f'(\zeta) \\[1em]
2Ax+B &= \frac{(b - a) \cdot f'(\zeta)} {b - a}
\end{align}$$

How to proceed further?

asked in Calculus by Veteran (20.2k points)  
edited by | 219 views

1 Answer

+4 votes
Best answer

Given  f(x) = Ax+ Bx +C, Domain  = [a,b]

This function is  polynomial function,so it is continuous & differentiable in its domain [a,b].

Hence LMVT(Lagrange's Mean Value Theorem) is applicable here.

Let there exists a  'ζ'  such that a < ζ < b, So from LMVT ,we have,

f(b) - f(a) /(b-a)  = f'(ζ)  => [(Ab+Bb +C) - (Aa+Ba +C) ]/(b-a)  = 2Aζ + B

=> [A(b2 - a2) + B(b-a)]/(b-a)  = 2Aζ + B

Since b & a are not equal so cancelling (b-a) from Numerator & denominator,we get,

=>A(b+a) + B  = 2Aζ + B  => ζ = (b+a)/2

Hence,  ζ = (b+a)/2

 

answered by Loyal (3.4k points)  
selected by

Related questions

+3 votes
1 answer
1
asked in Calculus by pC Veteran (20.2k points)   | 132 views
+1 vote
1 answer
2
Top Users Feb 2017
  1. Arjun

    5386 Points

  2. Bikram

    4230 Points

  3. Habibkhan

    3952 Points

  4. Aboveallplayer

    3086 Points

  5. Debashish Deka

    2564 Points

  6. sriv_shubham

    2318 Points

  7. Smriti012

    2236 Points

  8. Arnabi

    2008 Points

  9. mcjoshi

    1696 Points

  10. sh!va

    1684 Points

Monthly Topper: Rs. 500 gift card

20,863 questions
26,021 answers
59,689 comments
22,131 users