Mark the correct alternative in the following:
Let $f(x)=x^{3}+a x^{2}+b x+5 \sin ^{2} x$ be an increasing function on the set $R$. Then, $a$ and $b$ satisfy.
A. $a^{2}-3 b-15>0$
B. $a^{2}-3 b+15>0$
C. $a^{2}-3 b+15<0$
D. $a>0$ and $b>0$
Formula:- (i) $a x^{2}+b x+c>0$ for all $x \Rightarrow a>0$ and $b^{2}-4 a c<0$
(ii) $a x^{2}+b x+c<0$ for all $x \Rightarrow a<0$ and $b^{2}-4 a c<0$
(iii) The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$
Given:-
$f(x)=x^{3}+a x^{2}+b x+5 \sin ^{2} x$
$\mathrm{d}\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{dx}}\right)=3 \mathrm{x}^{2}+2 \mathrm{ax}+\mathrm{b}+5 \sin ^{2} \mathrm{x}=\mathrm{f}(\mathrm{x})$
For increasing function $f^{\prime}(x)>0$
$3 x^{2}+2 a x+b+5 \sin 2 x>0$
Then
$3 x^{2}+2 a x+b-5<0$
And $b^{2}-4 a c<0$
$\Rightarrow 4 a^{2}-12(b-5)<0$
$\Rightarrow a^{2}-3 b+15<0$
$\Rightarrow a^{2}-3 b+15<0$
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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,