Question:
If $y=\frac{x}{x+1}+\frac{x+1}{x},$ then $\frac{d^{2}y}{dx^{2}} $ at $x=1$ is equal to
If $y=\frac{x}{x+1}+\frac{x+1}{x},$ then $\frac{d^{2}y}{dx^{2}} $ at $x=1$ is equal to
Updated On: Jun 6, 2022
- $\frac{7}{4}$
- $\frac{7}{8}$
- $\frac{1}{4}$
- $\frac{-7}{8}$
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
$y=\frac{x}{x+1}+\frac{x+1}{x}$
$\Rightarrow \frac{d y}{d x}=\frac{(1)(x+1)-x(1)}{(x+1)^{2}}$
$+\frac{(1)(x)-(x+1) \cdot 1}{x^{2}}$
$=\frac{1}{(x+1)^{2}}-\frac{1}{x^{2}}$
$\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{-2}{(1+x)^{3}}+\frac{2}{x^{3}}$
On putting $x=1$, we get
$\left(\frac{d^{2} y}{d x^{2}}\right)_{x=1} =\frac{-2}{(1+1)^{3}}+\frac{2}{(1)^{3}}$
$=\frac{-2}{8}+2 $
$=-\frac{1}{4}+2=\frac{7}{4} $
$\Rightarrow \frac{d y}{d x}=\frac{(1)(x+1)-x(1)}{(x+1)^{2}}$
$+\frac{(1)(x)-(x+1) \cdot 1}{x^{2}}$
$=\frac{1}{(x+1)^{2}}-\frac{1}{x^{2}}$
$\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{-2}{(1+x)^{3}}+\frac{2}{x^{3}}$
On putting $x=1$, we get
$\left(\frac{d^{2} y}{d x^{2}}\right)_{x=1} =\frac{-2}{(1+1)^{3}}+\frac{2}{(1)^{3}}$
$=\frac{-2}{8}+2 $
$=-\frac{1}{4}+2=\frac{7}{4} $
Was this answer helpful?
0
0
Top Questions on Differentiability
If \( y = e^{{2}\log_e t} \) and \( x = \log_3(e^{t^2}) \), then \( \frac{dy}{dx} \) is equal to:
- CUET (UG) - 2024
- Mathematics
- Differentiability
- If \[y = \frac{\left(\sqrt{x} + 1\right)\left(x^2 - \sqrt{x}\right)}{x\sqrt{x} + x + \sqrt{x}} + \frac{1}{15}(3\cos^2 x - 5)\cos^3 x,\]then $96y'\left(\frac{\pi}{6}\right)$ is equal to:
- JEE Main - 2024
- Mathematics
- Differentiability
- Let \( g(x) \) be a linear function and \[ f(x) = \begin{cases} g(x), & x \leq 0 \\ \left( \frac{1 + x}{2 + x} \right)^{\frac{1}{x}}, & x > 0 \end{cases} \] is continuous at \( x = 0 \). If \( f'(1) = f(-1) \), then the value of \( g(3) \) is
- JEE Main - 2024
- Mathematics
- Differentiability
- Consider the function \( f : (0, \infty) \rightarrow \mathbb{R} \) defined by \(f(x) = e^{-| \log_e x |}.\)If \( m \) and \( n \) be respectively the number of points at which \( f \) is not continuous and \( f \) is not differentiable, then \( m + n \) is
- JEE Main - 2024
- Mathematics
- Differentiability
- The function f(x) = |x| + |1 − x| is:
- CUET (UG) - 2024
- Mathematics
- Differentiability
View More Questions
Questions Asked in KEAM exam
- A Spherical ball is subjected to a pressure of 100 atmosphere. If the bulk modulus of the ball is 1011 N/m2,then change in the volume is:
- KEAM - 2023
- mechanical properties of solids
- Two identical solid spheres,each of radius 10cm,are kept in contact. If the mass of inertia of this system about the tangent passing through the point of contact is 0. Then mass of each sphere is:
- KEAM - 2023
- System of Particles & Rotational Motion
- The value of a(≠0) for which the equation \(\frac{1}{2}(x-2)^2+1=\sin(\frac{a}{x})\) holds is/are
- KEAM - 2023
- Trigonometric Functions
- \(∫xlog(1+x^2)dx=\)?
- KEAM - 2023
- Integration by Parts
- A uniform thin rod of mass 3kg has a length of 1m. If a point mass of 1kg is attached to it at a distance of 40cm from its center,the center of mass shifts by a distance of:
- KEAM - 2023
- System of Particles & Rotational Motion
View More Questions
Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.