Mark the correct alternative in the following:
The function $f(x)=\log _{e}\left(x^{3}+\sqrt{x^{6}+1}\right)$ is of the following types:
A. even and increasing
B. odd and increasing
C. even and decreasing
D. odd and decreasing
Formula:- (i) if $f(-x)=f(x)$ then function is even
(ii) if $f(-x)=-f(x)$ then function is odd
(iii) The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$
Given:-
$f(x)=\log _{e}\left(x^{3}+\sqrt{x^{6}+1}\right)$
$\mathrm{d}\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{dx}}\right)=\frac{1}{\mathrm{x}^{3}\left(\mathrm{x}^{6}+1\right)^{\frac{1}{2}}}\left(3 \mathrm{x}^{2}+\frac{6 \mathrm{x}^{5}}{2\left(\mathrm{x}^{6}+1\right)^{\frac{1}{2}}}\right)$
$f^{\prime}(x)>0$
hence function is increasing function
$f(-x)=-\log \left(\log _{e}\left(x^{3}+\sqrt{x^{6}+1}\right)\right.$
$\Rightarrow f(-x)=-f(x)$ is odd function
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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,