Question:
The function f : R → R is defined by f(x) = cos2 x + sin4 x. Then, f(R) =
(a) [3/4, 1)
(b) (3/4, 1]
(c) [3/4, 1]
(d) (3/4, 1)
Solution:
(c) (3/4, 1)
Given:
f(x) = cos2x + sin4x
$\Rightarrow f(x)=1-\sin ^{2} x+\sin ^{4} x$
$\Rightarrow f(x)=\left(\sin ^{2} x-\frac{1}{2}\right)^{2}+\frac{3}{4}$
The minimum value of $f(x)$ is $\frac{3}{4}$.
Also,
$\sin ^{2} x \leq 1$
$\Rightarrow \sin ^{2} x-\frac{1}{2} \leq \frac{1}{2}$
$\Rightarrow\left(\sin ^{2} x-\frac{1}{2}\right)^{2} \leq \frac{1}{4}$
$\Rightarrow\left(\sin ^{2} x-\frac{1}{2}\right)^{2}+\frac{3}{4} \leq \frac{1}{4}+\frac{3}{4}$
$\Rightarrow f(x) \leq 1$
The maximum value of $f(x)$ is 1 .
∴ f(R) = (3/4, 1)
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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
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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