Find the absolute maximum and minimum values of the function of given by
$f(x)=\cos ^{2} x+\sin x, x \in[0, \pi]$
Given : $f(x)=\cos ^{2} x+\sin x$
$\Rightarrow f^{\prime}(x)=2 \cos x(-\sin x)+\cos x=-2 \sin x \cos x+\cos x$
For a local maximum or a local minimum, we must have
$f^{\prime}(x)=0$
$\Rightarrow-2 \sin x \cos x+\cos x=0$
$\Rightarrow \cos x(2 \sin x-1)=0$
$\Rightarrow \sin x=\frac{1}{2}$ or $\cos x=0$
$\Rightarrow x=\frac{\pi}{6}$ or $\frac{\pi}{2}$ $[\because x \in(0, \pi)]$
Thus, the critical points of $f$ are $0, \frac{\pi}{6}, \frac{\pi}{2}$ and $\pi$.
Now,
$f(0)=\cos ^{2}(0)+\sin (0)=1$
$f\left(\frac{\pi}{6}\right)=\cos ^{2}\left(\frac{\pi}{6}\right)+\sin \left(\frac{\pi}{6}\right)=\frac{5}{4}$
$f\left(\frac{\pi}{2}\right)=\cos ^{2}\left(\frac{\pi}{2}\right)+\sin \left(\frac{\pi}{2}\right)=1$
$f(\pi)=\cos ^{2}(\pi)+\sin (\pi)=1$
Hence, the absolute maximum value when $x=\frac{\pi}{6}$ is $\frac{5}{4}$ and the absolute minimum value when $x=0, \frac{\pi}{2}, \pi$ is $1 .$
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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,