For the function $\mathrm{f}(\mathrm{x})=x+\frac{1}{x}$
(a) $x=1$ is a point of maximum
(b) $x=-1$ is a point of minimum
(c) maximum value $>$ minimum value
(d) maximum value $<$ minimum value
(d) maximum value $<$ minimum value
Given : $f(x)=x+\frac{1}{x}$
$\Rightarrow f^{\prime}(x)=1-\frac{1}{x^{2}}$
For a local maxima or a local minima, we must have
$f^{\prime}(x)=0$
$\Rightarrow 1-\frac{1}{x^{2}}=0$
$\Rightarrow x^{2}-1=0$
$\Rightarrow x^{2}=1$
$\Rightarrow x=\pm 1$
Now,
$f^{\prime \prime}(x)=\frac{2}{x^{3}}$
$\Rightarrow f^{\prime \prime}(1)=\frac{2}{1}=2>0$
So, $x=1$ is a local minima.
Also,
$f^{\prime \prime}(-1)=-2<0$
So, $x=-1$ is a local maxima.
The local minimum value is given by
$f(1)=2$
The local maximum value is given by
$f(-1)=-2$
$\therefore$ Maximum value $<$ Minimum value
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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,