Chat with tutor

Ask Questions, Get Answers


Verify Lagrange's theorem for the following function;\[\]$f(x)=x^{3}-5x^{2}-3x\;[1,3]$

Note: This is part 5th of a 5 part question, split as 5 separate questions here.

1 Answer

  • Lagrange's Mean Value Theorem :
  • Let $f(x) $ be a real valued function that satisfies the following conditions.
  • (i) $f(x)$ is continuous on the closed interval $[a,b]$
  • (ii) $f(x)$ is differentiable in the open interval $(a,b)$
  • (iii) $f(a)=f(b)$
  • Then there exists atleast one value $c \in (a,b)$ such that $f'(c)=0$
$f(x)=x^3-5x^2-3x \qquad[1,3]$
Step 1:
$f(x)$ is a polynomial function .
It is continuous in $[1,3]$ and differentiable in $(1,3) $
The conditions for Lagranges Theorem are satisfied.
Step 2:
To find $c \in (1,3)$
Such that $f'(c)=\large\frac{f(3)-f(1)}{3-1}$
$f'{x}=3x^2-10 x -3$
$3c^2-10 c-3=-10$
$3c^2-10 c+7=0 \qquad c=\large\frac{10 \pm \sqrt {100-84}}{6}=\frac{10 \pm 4}{6}=\frac{14}{6},$$1$
$\large\frac{14}{6} =\frac{7}{3} \in $$(1,3)$ is the required 'c'
Lagrange's Theorem is verified
Help Clay6 to be free
Clay6 needs your help to survive. We have roughly 7 lakh students visiting us monthly. We want to keep our services free and improve with prompt help and advanced solutions by adding more teachers and infrastructure.

A small donation from you will help us reach that goal faster. Talk to your parents, teachers and school and spread the word about clay6. You can pay online or send a cheque.

Thanks for your support.
Please choose your payment mode to continue
Home Ask Homework Questions
Your payment for is successful.
Clay6 tutors use Telegram* chat app to help students with their questions and doubts.
Do you have the Telegram chat app installed?
Already installed Install now
*Telegram is a chat app like WhatsApp / Facebook Messenger / Skype.