Question:
Maximum slope of the curve $y=-x^{3}+3 x^{2}+9 x-27$ is
(a) 0
(b) 12
(c) 16
(d) 32
Solution:
The given curve is $y=-x^{3}+3 x^{2}+9 x-27$.
Slope of the curve, $m=\frac{d y}{d x}$
$\therefore m=\frac{d y}{d x}=-3 x^{2}+6 x+9$
$\Rightarrow \frac{d m}{d x}=-6 x+6$
For maxima or minima,
$\frac{d m}{d x}=0$
$\Rightarrow-6 x+6=0$
$\Rightarrow x=1$
Now,
$\frac{d^{2} m}{d x^{2}}=-6<0$
⇒ x = 1 is the point of local maximum
So, the slope of the given curve is maximum when x = 1.
∴ Maximum value of the slope, m
$=-3 \times(1)^{2}+6 \times 1+9 \quad\left(m=-3 x^{2}+6 x+9\right)$
$=-3+6+9$
$=12$
Thus, the maximum slope of the curve $y=-x^{3}+3 x^{2}+9 x-27$ is 12 .
Hence, the correct answer is option (b).
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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
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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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