How should we choose two numbers, each greater than or equal to -2 whose sum______________ so that the sum of the first and the cube of the second is minimum?
Let the two numbers be $x$ and $y$. Then,
$x, y>-2$ and $x+y=\frac{1}{2}$ .....(1)
Now,
$z=x+y^{3}$
$\Rightarrow z=x+\left(\frac{1}{2}-x\right)^{3}$ [From eq. (1)]
$\Rightarrow \frac{d z}{d x}=1+3\left(\frac{1}{2}-x\right)^{2}$
For maximum or minimum values of $z$, we must have
$\frac{d z}{d x}=0$
$\Rightarrow 1+3\left(\frac{1}{2}-x\right)^{2}=0$
$\Rightarrow\left(\frac{1}{2}-x\right)^{2}=\frac{1}{3}$
$\Rightarrow\left(\frac{1}{2}-x\right)=\pm \frac{1}{\sqrt{3}}$
$\Rightarrow x=\frac{1}{2} \pm \frac{1}{\sqrt{3}}$
$\frac{d^{2} z}{d x^{2}}=6\left(\frac{1}{2}-x\right)$
$\Rightarrow \frac{d^{2} z}{d x^{2}}=3-6 x$
At $x=\frac{1}{2} \pm \frac{1}{\sqrt{3}}:$
$\frac{d^{2} z}{d x^{2}}=3-6\left(\frac{1}{2}+\frac{1}{\sqrt{3}}\right)$
$\Rightarrow \frac{-6}{\sqrt{3}}<0$
Thus, $z$ is maximum when $x=\frac{1}{2}+\frac{1}{\sqrt{3}}$.
At $x=\frac{1}{2}-\frac{1}{\sqrt{3}}:$
$\frac{d^{2} z}{d x^{2}}=3-6\left(\frac{1}{2}-\frac{1}{\sqrt{3}}\right)$
$\Rightarrow \frac{6}{\sqrt{3}}>0$
Thus, $z$ is minimum when $x=\frac{1}{2}-\frac{1}{\sqrt{3}}$.
$x+y=\frac{1}{2}$
Substituting the value of $x$ in eq. (1), we get
$y=-\frac{1}{2}+\frac{1}{\sqrt{3}}+\frac{1}{2}$
$y=\frac{1}{\sqrt{3}}$
So, the required two numbers are $\left(\frac{1}{2}-\frac{1}{\sqrt{3}}\right)$ and $\frac{1}{\sqrt{3}}$.
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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,