If $f(x)=x|x|$, then $f(-1)=$____________
$|x|= \begin{cases}x, & x \geq 0 \\ -x, & x<0\end{cases}$
$\therefore f(x)=x|x|= \begin{cases}x^{2}, & x \geq 0 \\ -x^{2}, & x<0\end{cases}$
Now,
$L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{f(-1-h)-f(-1)}{-h}$
$\Rightarrow L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-(-1-h)^{2}-\left[-(-1)^{2}\right]}{-h}$
$\Rightarrow L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-\left(1+2 h+h^{2}\right)+1}{-h}$
$\Rightarrow L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-\left(2 h+h^{2}\right)}{-h}$
$\Rightarrow L f^{\prime}(-1)=\lim _{h \rightarrow 0}(2+h)$
$\Rightarrow L f^{\prime}(-1)=2+0=2$
Also,
$R f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{f(-1+h)-f(-1)}{h}$
$\Rightarrow R f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-(-1+h)^{2}-\left[-(-1)^{2}\right]}{h}$
$\Rightarrow R f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-\left(1-2 h+h^{2}\right)+1}{h}$
$\Rightarrow R f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-\left(-2 h+h^{2}\right)}{h}$
$\Rightarrow R f^{\prime}(-1)=\lim _{h \rightarrow 0}(2-h)$
$\Rightarrow R f^{\prime}(-1)=2-0=2$
So, $L f^{\prime}(-1)=R f^{\prime}(-1)=2$
$\therefore f^{\prime}(-1)=2$
If $f(x)=x|x|$, then $f(-1)=$ __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,