Show that $f(x)=\cos x$ is a decreasing function on $(0, \pi)$, increasing in $(-\pi, 0)$ and neither increasing nor
decreasing in $(-\pi, \pi)$.
Given:- Function $f(x)=\cos x$
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.
$f(x)=\cos x$
$\Rightarrow f(x)=\frac{d}{d x}(\cos x)$
$\Rightarrow f^{\prime}(x)=-\sin x$
Taking different region from 0 to $2 \pi$
a) let $x \in(0, \pi)$
$\Rightarrow \sin (x)>0$
$\Rightarrow-\sin x<0$
$\Rightarrow f^{\prime}(x)<0$
Thus $f(x)$ is decreasing in $(0, \pi)$
b) let $x \in(-\pi, 0)$
$\Rightarrow \sin (x)<0$
$\Rightarrow-\sin x>0$
$\Rightarrow f^{\prime}(x)>0$
Thus $f(x)$ is increasing in $(-\pi, 0)$
Therefore, from above condition we find that
$\Rightarrow f(x)$ is decreasing in $(0, \pi)$ and increasing in $(-\pi, 0)$
Hence, condition for $f(x)$ neither increasing nor decreasing in $(-\pi, \pi)$
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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,