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Is following function Satisfy Lagrange's Mean Value Theroem?
f(x) = | x+2 |  in [-2, 0]

asked in Calculus | 323 views
@habib..is it correct?

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.

answered by Veteran (14.8k points) 15 152 318
selected
@Akriti
why are we checking continuity at -2, -1 and 0 only why not at other values in between 0 and -2 including bounderies. there can be other values also
@vijay,there are numerous decimal values between -2 and 0.for howw many values will u check??and even if u check,you will get that the function is continous as well as differentiable.

in general,u just need to find at the boundaries and in between numbers in the set like above.
Can you please generalize it? so that i can follow the approach in furture.
suppose if interval is given as [0,6] then should i check differerntiability and continuity at 0,1,2,3,4,5, and 6?
continuity at 0,1,2,3,4,5,6

but differentiability at 1,2,3,4,5 because for diff ,it is open bracket ,not to find on boundaries.
yes correct! thank you :)
welcome :-)
@Akriti sood
only 2 conditions for lagranges mean value?...or there is something more?
if nt  , then whats the difference between it,rolle's theorem and cauchy?

for roll's theorem,there is one more condition,

f(a) = f(b) should be satisfied

whereas cauchy is applied on two functions defined on same domain

1) f and g should be continous in [a,b]

2) f and g should be derivable in ]a,b[

3) g'(x) !=0 for all x in ]a,b[

u r missing one very important point in all of these...continuity and differentiability is fine..but to say whether lagranges theorem will be applicble or not u need to find a c(mean value) belonging to (a,b)..isnt it?
and same with rolles theorem and cauchy..

but first the conditions should be satisfied..only whne these conditions are satisfied then only we find that c value in [z,b]
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 (12.5k points) 12 52 155

+1 vote