Question:
$\mathrm{f}(\mathrm{x})=x \sqrt{1-x}, x>0$
Solution:
Given : $f(x)=x \sqrt{1-x}$
$\Rightarrow f^{\prime}(x)=\sqrt{1-x}-\frac{x}{2 \sqrt{1-x}}=\frac{2-3 x}{2 \sqrt{1-x}}$
For the local maxima or minima, we must have
$f^{\prime}(x)=0$
$\Rightarrow \frac{2-3 x}{2 \sqrt{1-x}}=0$
$\Rightarrow x=\frac{2}{3}$
Since, $f^{\prime}(x)$ changes from positive to negative when $x$ increases through $\frac{2}{3}, x=\frac{2}{3}$ is a point of maxima.
The local maximum value of $f(x)$ at $x=\frac{2}{3}$ is given by
$\frac{2}{3} \sqrt{1-\frac{2}{3}}=\frac{2}{3 \sqrt{3}}=\frac{2 \sqrt{3}}{9}$
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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
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- Dual Nature of Radiation and Matter
- Atoms
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