Show that the function $\mathrm{f}(\mathrm{x})=\sin \left(2 \mathrm{x}+\frac{\pi}{4}\right)$ is decreasing on $\left(\frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$
Given:- Function $\mathrm{f}(\mathrm{x})=\sin \left(2 \mathrm{x}+\frac{\pi}{4}\right)$
Theorem:- Let $\mathrm{f}$ be a differentiable real function defined on an open interval $(\mathrm{a}, \mathrm{b})$.
(i) If $f^{\prime}(x)>0$ for $a l l x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$
(ii) If $f^{\prime}(x)<0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$
Algorithm:-
(i) Obtain the function and put it equal to $f(x)$
(ii) Find $f^{\prime}(x)$
(iii) Put $f^{\prime}(x)>0$ and solve this inequation.
For the value of $x$ obtained in (ii) $f(x)$ is increasing and for remaining points in its domain it is decreasing.
Here we have,
$\mathrm{f}(\mathrm{x})=\sin \left(2 \mathrm{x}+\frac{\pi}{4}\right)$
$\Rightarrow \mathrm{f}(\mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}\left\{\sin \left(2 \mathrm{x}+\frac{\pi}{4}\right)\right\}$
$\Rightarrow \mathrm{f}(\mathrm{x})=\cos \left(2 \mathrm{x}+\frac{\pi}{4}\right) \times 2$
$\Rightarrow \mathrm{f}(\mathrm{x})=2 \cos \left(2 \mathrm{x}+\frac{\pi}{4}\right)$
Now, as given
$x \in\left(\frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$
$\Rightarrow \frac{3 \pi}{8} $\Rightarrow \frac{3 \pi}{4}<2 x<\frac{5 \pi}{4}$ $\Rightarrow \pi<2 x+\frac{\pi}{4}<\frac{3 \pi}{2}$; as here $2 x+\frac{\pi}{4}$ lies in $3^{\text {rd }}$ quadrant $\Rightarrow \cos \left(2 x+\frac{\pi}{4}\right)<0$ $\Rightarrow 2 \cos \left(2 x+\frac{\pi}{4}\right)<0$ $\Rightarrow f^{\prime}(x)<0$ hence, Condition for $f(x)$ to be decreasing Thus $f(x)$ is decreasing on interval $\left(\frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$
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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,