Question:
(i) $f(x)=4 x-\frac{x^{2}}{2}$ in $[-2,4,5]$
(ii) $f(x)=(x-1)^{2}+3$ in $[-3,1]$
(iii) $f(x)=3 x 4-8 x^{3}+12 x^{2}-48 x+25$ in $[0,3]$
(iv) $f(x)=(x-2) \sqrt{x-1}$ in $[1,9]$
Solution:
(i)
Given: $f(x)=4 x-\frac{x^{2}}{2}$
$\Rightarrow f^{\prime}(x)=4-x$
For a local maximum or a local minimum, we must have
$f^{\prime}(x)=0$
$\Rightarrow 4-x=0$
$\Rightarrow x=4$
Thus, the critical points of $f$ are $-2,4$ and $4.5$.
Now,
$f(-2)=4(-2)-\frac{(-2)^{2}}{2}=-8-2=-10$
$f(4)=4(4)-\frac{(4)^{2}}{2}=16-8=8$
$f(4.5)=4(4.5)-\frac{(4.5)^{2}}{2}=18-10.125=7.875$
Hence, the absolute maximum value when $x=4$ is 8 and the absolute minimum value when $x=-2$ is $-10$.
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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,
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