Show that f (x) = 2x + cot-1 x + log [√(1 + x2) – x] is increasing in R.
Given,
f (x) = 2x + cot-1 x + log [√(1 + x2) – x]
Differentiating both sides w.r.t. x, we get
$f^{\prime}(x)=2-\frac{1}{1+x^{2}}+\frac{1}{\sqrt{1+x^{2}}-x} \times \frac{d}{d x}\left(\sqrt{1+x^{2}}-x\right)$
$=2-\frac{1}{1+x^{2}}+\frac{\left(\frac{1}{2 \sqrt{1+x^{2}}} \times(2 x-1)\right)}{\sqrt{1+x^{2}}-x}$
$=2-\frac{1}{1+x^{2}}+\frac{x-\sqrt{1+x^{2}}}{\sqrt{1+x^{2}}\left(\sqrt{1+x^{2}-x}\right)}$
$=2-\frac{1}{1+x^{2}}-\frac{\left(\sqrt{1+x^{2}}-x\right)}{\sqrt{1+x^{2}}\left(\sqrt{1+x^{2}}-x\right)}$
$=2-\frac{1}{1+x^{2}}-\frac{1}{\sqrt{1+x^{2}}}$
For increasing function, $f^{\prime}(x) \geq 0$
$2-\frac{1}{1+x^{2}}-\frac{1}{\sqrt{1+x^{2}}} \geq 0$
$\frac{2\left(1+x^{2}\right)-1+\sqrt{1+x^{2}}}{\left(1+x^{2}\right)} \geq 0 \Rightarrow 2+2 x^{2}-1+\sqrt{1+x^{2}} \geq 0$
$2 x^{2}+1+\sqrt{1+x^{2}} \geq 0 \Rightarrow 2 x^{2}+1 \geq-\sqrt{1+x^{2}}$
On squaring both the sides, we get
4x4 + 1 + 4x2 ≥ 1 + x2
4x4 + 4x2 – x2 ≥ 0
4x4 + 3x2 ≥ 0
x2(4x2 + 3) ≥ 0
The above is true for any value of x ∈ R.
Therefore, the given function is an increasing function over R.
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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,