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Mathematics Question on Statistics

If f(x)=xm f(x) = x^m, where mm is a positive integer then the value of mm for which f′(α+β)=f′(α)+f′(β)f'(\alpha + \beta) = f'(\alpha ) + f'(\beta) for all α,β>0 \alpha , \beta > 0 is

A

3

B

1

C

4

D

2

Answer

2

Explanation

Solution

f(x)=xmf\left(x\right)=x^{m} or f′(x)=m(x)m−1f'\left(x\right)=m\left(x\right)^{m-1}
Now, f′(α+β)=m(α+β)m−1f'\left(\alpha+\beta\right)=m\left(\alpha+\beta\right)^{m-1}
f′(α)=mαm−1,f′(β)=mβm−1f'\left(\alpha\right)=m \alpha^{m-1}, f'\left(\beta\right) =m \beta^{m-1}
m(α+β)m−1=m(αm−1+βm−1)m\left(\alpha+\beta\right)^{m-1} =m\left(\alpha^{m-1}+\beta^{m-1}\right)
(α+β)m−1=αm−1+βm−1\left(\alpha+\beta\right)^{m-1} =\alpha^{m-1}+\beta^{m-1}