Question:
Let $f(x)=x^{3}+3 x^{2}-9 x+2$. Then, $f(x)$ has
(a) a maximum at $x=1$
(b) a minimum at $x=1$
(c) neither a maximum nor a minimum at $x=-3$
(d) none of these
Solution:
(b) a minimum at $x=1$
Given: $f(x)=x^{3}+3 x^{2}-9 x+2$
$\Rightarrow f^{\prime}(x)=3 x^{2}+6 x-9$
For a local maxima or a local minima, we must have
$f^{\prime}(x)=0$
$\Rightarrow 3 x^{2}+6 x-9=0$
$\Rightarrow x^{2}+2 x-3=0$
$\Rightarrow(x+3)(x-1)=0$
$\Rightarrow x=-3,1$
Now,
$f^{\prime \prime}(x)=6 x+6$
$\Rightarrow f^{\prime \prime}(1)=6+6=12>0$
So, $x=1$ is a local minima.
Also,
$f^{\prime \prime}(-3)=-18+6=-12<0$
So, $x=-3$ is a local maxima.
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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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