The minimum value of $f(x)=\sin x$ in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ isThe minimum value of $f(x)=\sin x$ in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ is_________
The given function is $f(x)=\sin x, x \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$.
$f(x)=\sin x$
Differentiating both sides with respect to x, we get
$f^{\prime}(x)=\cos x$
For maxima or minima,
$f^{\prime}(x)=0$
$\Rightarrow \cos x=0$
$\Rightarrow x=-\frac{\pi}{2}$ or $x=\frac{\pi}{2}$, for all $x \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$
Now,
$f^{\prime \prime}(x)=-\sin x$
At $x=\frac{\pi}{2}$, we have
$f^{\prime \prime}\left(\frac{\pi}{2}\right)=-\sin \frac{\pi}{2}=-1<0$
So, $x=\frac{\pi}{2}$ is the point of local maximum of $f(x)$.
At $x=-\frac{\pi}{2}$, we have
$f^{\prime \prime}\left(-\frac{\pi}{2}\right)=-\sin \left(-\frac{\pi}{2}\right)=\sin \frac{\pi}{2}=1>0$ $[\sin (-\theta)=-\sin \theta]$
So, $x=-\frac{\pi}{2}$ is the point of local minimum of $f(x)$.
$\therefore$ Minimum value of $f(x)=f\left(-\frac{\pi}{2}\right)=\sin \left(-\frac{\pi}{2}\right)=-\sin \frac{\pi}{2}=-1$
Thus, the minimum value of $f(x)=\sin x$ in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ is $-1$.
The minimum value of $f(x)=\sin x$ in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ is ___−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,