GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
153 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 (14.6k points)   | 153 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 (12.5k 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 (11.5k points)  


Top Users Mar 2017
  1. rude

    4758 Points

  2. sh!va

    3014 Points

  3. Rahul Jain25

    2830 Points

  4. Kapil

    2636 Points

  5. Debashish Deka

    2450 Points

  6. 2018

    1514 Points

  7. Vignesh Sekar

    1422 Points

  8. Akriti sood

    1314 Points

  9. Bikram

    1286 Points

  10. Sanjay Sharma

    1076 Points

Monthly Topper: Rs. 500 gift card

21,484 questions
26,812 answers
61,056 comments
23,065 users