The maximum value of $\left(\frac{1}{x}\right)^{x}$ is
(a) $e$
(b) $e^{e}$
(c) $e^{1 / e}$
(d) $\left(\frac{1}{e}\right)^{1 / e}$
Let $f(x)=\left(\frac{1}{x}\right)^{x}$.
$f(x)=\left(\frac{1}{x}\right)^{x}$
$\Rightarrow \log f(x)=\log \left(\frac{1}{x}\right)^{x}$
$\Rightarrow \log f(x)=x \log \left(\frac{1}{x}\right)$
$\Rightarrow \log f(x)=-x \log x$
Differentiating both sides with respect to x, we get
$\frac{1}{f(x)} \times f^{\prime}(x)=-\left(x \times \frac{1}{x}+\log x \times 1\right)$
$\Rightarrow f^{\prime}(x)=-\left(\frac{1}{x}\right)^{x}(1+\log x)$ .....(1)
For maxima or minima,
$f^{\prime}(x)=0$
$\Rightarrow-\left(\frac{1}{x}\right)^{x}(1+\log x)=0$
$\Rightarrow 1+\log x=0 \quad\left[\left(\frac{1}{x}\right)^{x}>0\right]$
$\Rightarrow \log x=-1$
$\Rightarrow x=e^{-1}=\frac{1}{e}$
Now,
$f^{\prime \prime}(x)=-\left[\left(\frac{1}{x}\right)^{x} \times \frac{1}{x}+(1+\log x) \times\left\{-\left(\frac{1}{x}\right)^{x}(1+\log x)\right\}\right]$ [Using (1)]
$\Rightarrow f^{\prime \prime}\left(\frac{1}{e}\right)=-\left[(e)^{\frac{1}{e}} \times e+\left(1+\log \frac{1}{e}\right) \times\left\{-(e)^{\frac{1}{e}}\left(1+\log \frac{1}{e}\right)\right\}\right]$ $\left(\log \frac{1}{e}=\log e^{-1}=-1\right)$
$\Rightarrow f^{\prime \prime}\left(\frac{1}{e}\right)=-e^{\frac{1}{e}+1}-0=-e^{\frac{1}{e}+1}$
$\Rightarrow f^{\prime \prime}\left(\frac{1}{e}\right)<0$
So, $x=\frac{1}{e}$ is the point of local maximum of $f(x)$.
$\therefore$ Maximum value of $f(x)=\left(\frac{1}{\frac{1}{e}}\right)^{\frac{1}{e}}=e^{\frac{1}{e}}$
Thus, the maximum value of $\left(\frac{1}{x}\right)^{x}$ is $e^{\frac{1}{e}}$.
Hence, the correct answer is option (c).
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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,