Let f(x)=xm, m being a non-negative integer. The value of m so that the equality f'(a+b)=f'(a)+f'(b) is valid for all a,b>0 is
- 0
- 1
- 2
- 3
The Correct Option is A, C
Solution and Explanation
The correct answer is/are option(s) :
(A): 0
(C): 2
\(f(x)=x^m\Rightarrow f'(x)=mx^{m-1}\)
\(f'(a+b)=f'(a)+f'(b)\)
\(\Rightarrow n(a+b)^{m-1}=a^{m-1}+b^{m-1}\)
Also, for \(m=0, f(x)=1\)
So, \(f'(x)=0;f'(a+b)=0\)
\(So, f'(a+b)=f'(a)+f'(b)\)
Hence, there are two values of m. which are 0 and 2.
Top Questions on limits and derivatives
- If the value of the integral \[ \int_{-1}^{1} \frac{\cos \alpha x}{1 + 3^x} \, dx = \frac{2}{\pi}, \] then a value of \( \alpha \) is:
- JEE Main - 2024
- Mathematics
- limits and derivatives
- Let \[ f(x) = \int_{0}^{x} \left( t + \sin\left(1 - e^t\right) \right) \, dt, \, x \in \mathbb{R}. \] Then \[ \lim_{x \to 0} \frac{f(x)}{x^3} \] is equal to:
- JEE Main - 2024
- Mathematics
- limits and derivatives
- Let $I(x) = \int \frac{6}{\sin^2 x (1 - \cot x)^2} dx$. If $I(0) = 3$, then $I\left(\frac{\pi}{12}\right)$ is equal to:
- JEE Main - 2024
- Mathematics
- limits and derivatives
- Let \[ \int_{\log_e a}^{4} \frac{dx}{\sqrt{e^x - 1}} = \frac{\pi}{6}. \] Then \(e^\alpha\) and \(e^{-\alpha}\) are the roots of the equation:
- JEE Main - 2024
- Mathematics
- limits and derivatives
- The value of \(\int_{-\pi}^{\pi} \frac{2y(1 + \sin y)}{1 + \cos^2 y} \, dy\)
- JEE Main - 2024
- Mathematics
- limits and derivatives
Questions Asked in WBJEE exam
- ABC is an isosceles triangle with an inscribed circle with center O. Let P be the midpoint of BC. If AB=AC=15 and BC=10,then OP equals
- WBJEE - 2023
- Three Dimensional Geometry
- Let A=\(\begin{pmatrix} 0&0 &1 \\ 1&0 &0 \\ 0&0 &0 \end{pmatrix}\), B=\(\begin{pmatrix} 0&1 &0 \\ 0&0 &1 \\ 0&0 &0 \end{pmatrix}\) and P=\(\begin{pmatrix} 0&1 &0 \\ x&0 &0 \\ 0&0 &y \end{pmatrix}\) be an orthogonal matrix such that B=PAP-1 holds. Then
- WBJEE - 2023
- Matrices
Let α,β be the roots of the equation, ax2+bx+c=0.a,b,c are real and sn=αn+βn and \(\begin{vmatrix}3 &1+s_1 &1+s_2\\1+s_1&1+s_2 &1+s_3\\1+s_2&1+s_3 &1+s_4\end{vmatrix}=\frac{k(a+b+c)^2}{a^4}\) then k=
- WBJEE - 2023
- Complex Numbers and Quadratic Equations
- An electric dipole of dipole moment p is placed at the origin of the coordinate system along the z-axis. The amount of work required to move a charge 'q' from the point (a,0,0) to the point (0,0,a) is
- WBJEE - 2023
- Electric Dipole
- If \(\int \frac{dx}{(x+1)(x-2)(x-3)}=\frac{1}{k}log_e\left \{ \frac{|x-3|^3|x+1|}{(x-2)^4}\right \}+c\), then the value of k is
- WBJEE - 2023
- Integrals of Some Particular Functions
Concepts Used:
Limits And Derivatives
Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
Properties of Derivatives:
Read More: Limits and Derivatives