Find the intervals in which the following functions are increasing or decreasing.
$f(x)=\frac{1}{4} x^{4}+\frac{2}{3} x^{3}-\frac{5}{2} x^{2}-6 x+7$
Given:- Function $\mathrm{f}(\mathrm{x})=\frac{1}{4} \mathrm{x}^{4}+\frac{2}{3} \mathrm{x}^{3}-\frac{5}{2} \mathrm{x}^{2}-6 \mathrm{x}+7$
Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$.
(i) If $f^{\prime}(x)>0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$
(ii) If $f^{\prime}(x)<0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$
Algorithm:-
(i) Obtain the function and put it equal to $f(x)$
(ii) Find $f^{\prime}(x)$
(iii) Put $f^{\prime}(x)>0$ and solve this inequation.
For the value of $x$ obtained in (ii) $f(x)$ is increasing and for remaining points in its domain it is decreasing.
Here we have,
$f(x)=\frac{1}{4} x^{4}+\frac{2}{3} x^{3}-\frac{5}{2} x^{2}-6 x+7$
$\Rightarrow f(x)=\frac{d}{d x}\left(\frac{1}{4} x^{4}+\frac{2}{3} x^{3}-\frac{5}{2} x^{2}-6 x+7\right)$
$\Rightarrow f^{\prime}(x)=x^{3}+2 x^{2}-5 x-6$
For $f(x)$ lets find critical point, we must have
$\Rightarrow f^{\prime}(x)=0$
$\Rightarrow x^{3}+2 x^{2}-5 x-6=0$
$\Rightarrow(x+1)(x-2)(x+3)=0$
$\Rightarrow x=-1,2,-3$
clearly, $f^{\prime}(x)>0$ if $-3
and $f^{\prime}(x)<0$ if $x<-3$ and $-3 Thus, $f(x)$ increases on $(-3,-1) \cup(2, \infty)$ and $f(x)$ is decreasing on interval $(\infty,-3) \cup(-1,2)$
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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,