Question:
Show that for a ³ 1, f (x) = √3 sin x – cos x – 2ax + b is decreasing in R.
Solution:
Given,
f (x) = √3 sin x – cos x – 2ax + b, a ³ 1
On differentiating both sides w.r.t. x, we get
f’ (x) = √3 cos x + sin x – 2a
For increasing function, f’ (x) < 0
So, $\sqrt{3} \cos x+\sin x-2 a<0$
$2\left(\frac{\sqrt{3}}{2} \cos x+\frac{1}{2} \sin x\right)-2 a<0$
$\frac{\sqrt{3}}{2} \cos x+\frac{1}{2} \sin x-a<0$
$\left(\cos \frac{\pi}{6} \cos x+\sin \frac{\pi}{6} \sin x\right)-a<0$
$\cos \left(x-\frac{\pi}{6}\right)-a<0$
As $\cos x \in[-1,1]$ and $a \geq 1$
Hence, $f^{\prime}(x)<0$
Therefore, the given function is decreasing in R.
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- 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
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- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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