Discuss the applicability of Lagrange's mean value theorem for the function $f\left( x \right) = \left| x \right|$ on $\left[ { - 1,1} \right]$.
Last updated date: 24th Mar 2023
•
Total views: 306.6k
•
Views today: 2.84k
Answer
306.6k+ views
Hint: In this question, for applicability of Lagrange's Mean Value theorem on f(x) in interval $\left[ {a,b} \right]$, function must be continuous in $\left[ {a,b} \right]$ and differentiable in (a,b).
Complete step-by-step answer:
Now we define the function $f\left( x \right) = \left| x \right|$ on $\left[ { - 1,1} \right]$ as follows,
$
f\left( x \right) = - x,x \in \left[ { - 1,0} \right) \\
f\left( x \right) = x,x \in \left[ {0,1} \right] \\
$
Now let's examine continuity and differentiability of function at x=0 .
For continuity,
\[
{\text{Left hand limit,}}\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = \mathop {\lim }\limits_{x \to {0^ - }} \left( { - x} \right) = 0 \\
{\text{Right hand limit,}}\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {0^ + }} \left( x \right) = 0 \\
\]
Since, LHL=RHL, f(x) is continuous at x=0 and f(0)=0 .
For differentiability,
Left hand derivative
$
f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x - h} \right) - f\left( x \right)}}{{ - h}} \\
f'\left( 0 \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {0 - h} \right) - f\left( 0 \right)}}{{ - h}} \\
f'\left( 0 \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( { - h} \right) - 0}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \dfrac{{ - \left( { - h} \right) - 0}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \dfrac{h}{{ - h}} = - 1 \\
$
Right hand derivative
$
f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h} \\
f'\left( 0 \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {0 + h} \right) - f\left( 0 \right)}}{h} \\
f'\left( 0 \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( h \right) - 0}}{h} = \mathop {\lim }\limits_{h \to 0} \dfrac{h}{h} = 1 \\
$
Since, $LHD \ne RHD$ , f(x) is not differentiable at x=0
For applicability of Lagrange's Mean Value theorem on f(x) in interval $\left[ { - 1,1} \right]$ , Function must be continuous in $\left[ { - 1,1} \right]$ and differentiable in (1,1) .
But the function is not differentiable at x=0 .
So, Lagrange's Mean Value theorem is not applicable for f(x) in interval $\left[ { - 1,1} \right]$.
Note: Whenever we face such types of problems we use some important points. First we define the function on different intervals then check the continuity and differentiability of the function on different intervals. If function be continuous and differentiable on interval then Lagrange's mean value theorem be applicable.
Complete step-by-step answer:
Now we define the function $f\left( x \right) = \left| x \right|$ on $\left[ { - 1,1} \right]$ as follows,
$
f\left( x \right) = - x,x \in \left[ { - 1,0} \right) \\
f\left( x \right) = x,x \in \left[ {0,1} \right] \\
$
Now let's examine continuity and differentiability of function at x=0 .
For continuity,
\[
{\text{Left hand limit,}}\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = \mathop {\lim }\limits_{x \to {0^ - }} \left( { - x} \right) = 0 \\
{\text{Right hand limit,}}\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {0^ + }} \left( x \right) = 0 \\
\]
Since, LHL=RHL, f(x) is continuous at x=0 and f(0)=0 .
For differentiability,
Left hand derivative
$
f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x - h} \right) - f\left( x \right)}}{{ - h}} \\
f'\left( 0 \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {0 - h} \right) - f\left( 0 \right)}}{{ - h}} \\
f'\left( 0 \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( { - h} \right) - 0}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \dfrac{{ - \left( { - h} \right) - 0}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \dfrac{h}{{ - h}} = - 1 \\
$
Right hand derivative
$
f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h} \\
f'\left( 0 \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {0 + h} \right) - f\left( 0 \right)}}{h} \\
f'\left( 0 \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( h \right) - 0}}{h} = \mathop {\lim }\limits_{h \to 0} \dfrac{h}{h} = 1 \\
$
Since, $LHD \ne RHD$ , f(x) is not differentiable at x=0
For applicability of Lagrange's Mean Value theorem on f(x) in interval $\left[ { - 1,1} \right]$ , Function must be continuous in $\left[ { - 1,1} \right]$ and differentiable in (1,1) .
But the function is not differentiable at x=0 .
So, Lagrange's Mean Value theorem is not applicable for f(x) in interval $\left[ { - 1,1} \right]$.
Note: Whenever we face such types of problems we use some important points. First we define the function on different intervals then check the continuity and differentiability of the function on different intervals. If function be continuous and differentiable on interval then Lagrange's mean value theorem be applicable.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
