Find the maximum and minimum values, if any, of the following functions given by
(i) $f(x)=|x+2|-1$
(ii) $g(x)=-|x+1|+3$
(iii) $h(x)=\sin (2 x)+5$
(iv) $f(x)=|\sin 4 x+3|$
(v) $h(x)=x+1, x \in(-1,1)$
(i) $f(x)=|x+2|-1$
We know that $|x+2| \geq 0$ for every $x \in \mathbf{R}$.
Therefore, $f(x)=|x+2|-1 \geq-1$ for every $x \in \mathbf{R}$.
The minimum value of $f$ is attained when $|x+2|=0$.
$|x+2|=0$
$\Rightarrow x=-2$
$\therefore$ Minimum value of $f=f(-2)==|-2+2|-1=-1$
Hence, function f does not have a maximum value.
(ii) $g(x)=-|x+1|+3$
We know that $-|x+1| \leq 0$ for every $x \in \mathbf{R}$.
Therefore, $g(x)=-|x+1|+3 \leq 3$ for every $x \in \mathbf{R}$.
The maximum value of $g$ is attained when $|x+1|=0$.
$|x+1|=0$
$\Rightarrow x=-1$
$\therefore$ Maximum value of $a=a(-1)=-|-1+1|+3=3$
(iii) $h(x)=\sin 2 x+5$
We know that $-1 \leq \sin 2 x \leq 1$.
$\Rightarrow-1+5 \leq \sin 2 x+5 \leq 1+5$
$\Rightarrow 4 \leq \sin 2 x+5 \leq 6$
Hence, the maximum and minimum values of h are 6 and 4 respectively.
(iv) $f(x)=|\sin 4 x+3|$
We know that $-1 \leq \sin 4 x \leq 1$.
$\Rightarrow 2 \leq \sin 4 x+3 \leq 4$
$\Rightarrow 2 \leq|\sin 4 x+3| \leq 4$
Hence, the maximum and minimum values of f are 4 and 2 respectively.
(v) $h(x)=x+1, x \in(-1,1)$
Here, if a point $x_{0}$ is closest to $-1$, then we find $\frac{x_{0}}{2}+1>x_{0}+1$ for all $x_{0} \in(-1,1)$.
Also, if $x_{1}$ is closest to 1 , then $x_{1}+1<\frac{x_{1}+1}{2}+1$ for all $x_{1} \in(-1,1)$.
Hence, function h(x) has neither maximum nor minimum value in (−1, 1).
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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,