Question:
The least value of the function $\mathrm{f}(\mathrm{x})=x 3-18 x 2+96 x$ in the interval $[0,9]$ is
(a) 126
(b) 135
(c) 160
(d) 0
Solution:
(d) 0
Given: $f(x)=x^{3}-18 x^{2}+96 x$
$\Rightarrow f^{\prime}(x)=3 x^{2}-36 x+96$
For a local maxima or a local minima, we must have
$f^{\prime}(x)=0$
$\Rightarrow 3 x^{2}-36 x+96=0$
$\Rightarrow x^{2}-12 x+32=0$
$\Rightarrow(x-4)(x-8)=0$
$\Rightarrow x=4,8$
So,
$f(8)=(8)^{3}-18(8)^{2}+96(8)=512-1152+768=128$
$f(4)=(4)^{3}-18(4)^{2}+96(4)=64-288+384=160$
$f(0)=(0)^{3}-18(0)^{2}+96(0)=0$
$f(9)=(9)^{3}-18(9)^{2}+96(9)=729-1458+864=135$
Hence, 0 is the minimum value in the range $[0,9]$.
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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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- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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