Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function such that
$f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in R$
Then $f(2)$ equal :
8
$-2$
$-4$
30
Correct Option: , 2
$f(x)=x^{3}+x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3)$
$\Rightarrow f^{\prime}(x)=3 x^{2}+2 x f^{\prime}(1)+f^{\prime \prime}(x)$ $\ldots \ldots(1)$
$\Rightarrow f^{\prime \prime}(x)=6 x+2 f^{\prime}(1)$ $\ldots \ldots(2)$
$\Rightarrow f^{\prime \prime \prime}(x)=6$ $\ldots . .(3)$
put $x=1$ in equation (1):
$f^{\prime}(1)=3+2 f^{\prime}(1)+f^{\prime \prime}(2)$ $\ldots . .(4)$
put $x=2$ in equation (2):
$f^{\prime \prime}(2)=12+2 f^{\prime}(1)$ $\ldots . .(5)$
from equation (4) \& (5) :
$-3-f^{\prime}(1)=12+2 f^{\prime}(1)$
$\Rightarrow 3 f^{\prime}(1)=-15$
$\Rightarrow f^{\prime}(1)=-5 \Rightarrow f^{\prime \prime}(2)=2 \quad \ldots .(2)$
put $x=3$ in equation (3):
$f^{\prime \prime \prime}(3)=6$
$\therefore f(x)=x^{3}-5 x^{2}+2 x+6$
$f(2)=8-20+4+6=-2$
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All Study Material
- 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,