GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
137 views
Is following function Satisfy Lagrange's Mean Value Theroem?
f(x) = | x+2 |  in [-2, 0]

Detailed solution PLEASE!
asked in Calculus by Veteran (14.4k points)   | 137 views
Change your comment to answer @Akriti sood
@habib..is it correct?

2 Answers

+4 votes
Best answer

for lagrange's theorem,

1) it should be continous at [-2,0]

f(x) = |x+2| =-(x+2 ) x< -2

(x+2)   -2<= x< -1

(x+2)   -1< x < 0

(x+2)    x>0

        

so,for continouty check limit at -2  lim x ->-2- -(x+2) =0

rhs at -2------ lim x-> -2(x+2) =0

and f(-2) =0

hence,continous at x=-2

similarly,it is continous at -1 and 0

2) f(x) should be differentiable at x= ]-2,0[

for checking diferentibilty

lim x-> c- f(x) -f(c)/x-c =lim x->cf(x) -f(c)/x-c

which is same as f(x) for lhs and rhs of -1 is same.

it should  follow lagranje

 

 

answered by Veteran (11.1k points)  
selected by
yes...but only satisfying continuity and differentiabilty doesnt guarantee u that u could always find such c in (a,b) right?
u should check for c also before saying whether lagranges will be applicable or nt!!
@Sudhso
Akriti is correct here, we just need to check first 2 conditions to check if lagrange's theroem can be applied or not
@vijay means every continuous and differentiable fn for a range is applicable for lagrange's mean value theorem?
@sudsho,
yes,if given function is continuous and differentiable, there will be such C definitely.
lagranges theorem means that, if a curve is smooth beween [a,b] and no sharp edges  then there will be a point "c" in (a,b) such that slope of line joining f(a),f(b) is equal to slope of tangent drawn at point C.
+1 vote
you can observe by substituting -2,-1,0 in the given expression that its a straight line joining (-2,0) and (0,2) in the 2nd quadrant.
its a straightline so its differentiable and continous.
so lLagrange's MVT is applicable
answered by Veteran (11.4k points)  
Top Users Feb 2017
  1. Arjun

    5274 Points

  2. Bikram

    4230 Points

  3. Habibkhan

    3842 Points

  4. Aboveallplayer

    3086 Points

  5. Debashish Deka

    2378 Points

  6. sriv_shubham

    2308 Points

  7. Smriti012

    2236 Points

  8. Arnabi

    2008 Points

  9. sh!va

    1672 Points

  10. mcjoshi

    1640 Points

Monthly Topper: Rs. 500 gift card

20,846 questions
26,001 answers
59,649 comments
22,098 users