If f x - x n - n being a non-negative integer- then the values of n for which f - f - f - for all - -0 isA- 1B- 2C- 0D- 5

The correct options are B 2 C 0f(x)=x^n⇒ f'(x)=nx^n 1 f'(α+β)=f'(α)+f'(β) ⇒ n(α+β)^n 1=nα^n 1+nβ^n 1 ⇒ n=0 or (α+β)^n 1=α^n 1+β^n 1 (α+β)^n 1=α^n 1+β^n 1 This ...

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Standard XII

Mathematics

Monotonically Increasing Functions

If f x = x n ...

Question

If $f (x) = x^{n}, n$ being a non-negative integer, then the values of $n$ for which $f^{'} (α + β) = f^{'} (α) + f^{'} (β)$ for all $α, β > 0$ is

1

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Solution

The correct options are
B $2$
C $0$
$f (x) = x^{n} \Rightarrow f^{'} (x) = n x^{n - 1}$

$f^{'} (α + β) = f^{'} (α) + f^{'} (β)$
$\Rightarrow n (α + β)^{n - 1} = n α^{n - 1} + n β^{n - 1}$
$\Rightarrow n = 0$ or $(α + β)^{n - 1} = α^{n - 1} + β^{n - 1}$
$(α + β)^{n - 1} = α^{n - 1} + β^{n - 1}$
This equation is satisfied for $n - 1 = 1 \Rightarrow n = 2$

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If f(x)=xn,n being a non-negative integer, then the values of n for which f′(α+β)=f′(α)+f′(β) for all α,β>0 is

The correct options are B 2 C 0f(x)=xn⇒f′(x)=nxn−1 f′(α+β)=f′(α)+f′(β) ⇒n(α+β)n−1=nαn−1+nβn−1 ⇒n=0 or (α+β)n−1=αn−1+βn−1 (α+β)n−1=αn−1+βn−1 This equation is satisfied for n−1=1⇒n=2

If $f (x) = x^{n}, n$ being a non-negative integer, then the values of $n$ for which $f^{'} (α + β) = f^{'} (α) + f^{'} (β)$ for all $α, β > 0$ is