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.
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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