Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
Mean Value Theorem states that for a function $f:[a, b] \rightarrow \mathbf{R}$, if
(a) f is continuous on [a, b]
(b) f is differentiable on (a, b)
then, there exists some $c \in(a, b)$ such that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$
Therefore, Mean Value Theorem is not applicable to those functions that do not satisfy any of the two conditions of the hypothesis.
(i) $f(x)=[x]$ for $x \in[5,9]$
It is evident that the given function f (x) is not continuous at every integral point.
In particular, $f(x)$ is not continuous at $x=5$ and $x=9$
$\Rightarrow f(x)$ is not continuous in $[5,9] .$
The differentiability of f in (5, 9) is checked as follows.
Let $n$ be an integer such that $n \in(5,9)$
The left hand limit of $f$ at $x=n$ is,
$\lim _{h \rightarrow 0^{0}} \frac{f(n+h)-f(n)}{h}=\lim _{h \rightarrow 0^{-}} \frac{[n+h]-[n]}{h}=\lim _{h \rightarrow 0} \frac{n-1-n}{h}=\lim _{h \rightarrow 0} \frac{-1}{h}=\infty$
The right hand limit of $f$ at $x=n$ is,
$\lim _{h \rightarrow 0^{+}} \frac{f(n+h)-f(n)}{h}=\lim _{h \rightarrow 0^{+}} \frac{[n+h]-[n]}{h}=\lim _{h \rightarrow 0^{0}} \frac{n-n}{h}=\lim _{h \rightarrow 0^{+}} 0=0$
Since the left and right hand limits of f at x = n are not equal, f is not differentiable at x = n
∴f is not differentiable in (5, 9).
It is observed that f does not satisfy all the conditions of the hypothesis of Mean Value Theorem.
Hence, Mean Value Theorem is not applicable for $f(x)=[x]$ for $x \in[5,9]$.
(ii) $f(x)=[x]$ for $x \in[-2,2]$
It is evident that the given function f (x) is not continuous at every integral point.
In particular, f(x) is not continuous at x = −2 and x = 2
$\Rightarrow f(x)$ is not continuous in $[-2,2] .$
The left hand limit of $f$ at $x=n$ is,
$\lim _{h \rightarrow 0} \frac{f(n+h)-f(n)}{h}=\lim _{h \rightarrow 0} \frac{[n+h]-[n]}{h}=\lim _{h \rightarrow 0} \frac{n-1-n}{h}=\lim _{h \rightarrow 0} \frac{-1}{h}=\infty$
The right hand limit of $f$ at $x=n$ is,
$\lim _{h \rightarrow 0^{+}} \frac{f(n+h)-f(n)}{h}=\lim _{h \rightarrow 0^{+}} \frac{[n+h]-[n]}{h}=\lim _{h \rightarrow 0^{+}} \frac{n-n}{h}=\lim _{h \rightarrow 0^{+}} 0=0$
Since the left and right hand limits of f at x = n are not equal, f is not differentiable at x = n
∴f is not differentiable in (−2, 2).
It is observed that f does not satisfy all the conditions of the hypothesis of Mean Value Theorem.
Hence, Mean Value Theorem is not applicable for $f(x)=[x]$ for $x \in[-2,2]$.
(iii) $f(x)=x^{2}-1$ for $x \in[1,2]$
It is evident that f, being a polynomial function, is continuous in [1, 2] and is differentiable in (1, 2).
It is observed that f satisfies all the conditions of the hypothesis of Mean Value Theorem.
Hence, Mean Value Theorem is applicable for $f(x)=x^{2}-1$ for $x \in[1,2]$.
It can be proved as follows.
$f(1)=1^{2}-1=0, f(2)=2^{2}-1=3$
$\therefore \frac{f(b)-f(a)}{b-a}=\frac{f(2)-f(1)}{2-1}=\frac{3-0}{1}=3$
$f^{\prime}(x)=2 x$
$\therefore f^{\prime}(c)=3$
$\Rightarrow 2 c=3$
$\Rightarrow c=\frac{3}{2}=1.5$, where $1.5 \in[1,2]$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,