If f x -1- x - x 2- x 1000- then the value of f -1 isA- -50B- -100C- -500D- 500500

The correct option is C 500f(x)=1+x+x^2+x^3+x^4+ ⋯+x^999+x^1000 f'(x)=0+1+2x+3x^2+4x^3+ ⋯ +999x^998+1000x^999 f'( 1)=0+1 2+3 4+ ⋯+999 1000 ⇒ f'( 1)=(1 2)+(3 4 ...

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Byju's Answer

Standard XII

Mathematics

Vn Method

If f x =1+ x ...

Question

If $f (x) = 1 + x + x^{2} + \dots + x^{1000}$ , then the value of $f^{'} (- 1)$ is

- 50

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- 500

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- 100

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500500

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Solution

The correct option is B $- 500$
$f (x) = 1 + x + x^{2} + x^{3} + x^{4} + \dots + x^{999} + x^{1000}$
$f^{'} (x) = 0 + 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots + 999 x^{998} + 1000 x^{999}$
$f^{'} (- 1) = 0 + 1 - 2 + 3 - 4 + \dots + 999 - 1000$
$\Rightarrow f^{'} (- 1) = (1 - 2) + (3 - 4) + \dots + (999 - 1000)$
$\Rightarrow f^{'} (- 1) = (- 1) + (- 1) + (- 1) + \dots upto 500 times$
$\Rightarrow f^{'} (- 1) = - 500$

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If f(x)=1+x+x2+⋯+x1000, then the value of f′(−1) is

The correct option is B −500f(x)=1+x+x2+x3+x4+⋯+x999+x1000 f′(x)=0+1+2x+3x2+4x3+⋯+999x998+1000x999 f′(−1)=0+1−2+3−4+⋯+999−1000 ⇒f′(−1)=(1−2)+(3−4)+⋯+(999−1000) ⇒f′(−1)=(−1)+(−1)+(−1)+⋯ upto 500 times ⇒f′(−1)=−500

If $f (x) = 1 + x + x^{2} + \dots + x^{1000}$ , then the value of $f^{'} (- 1)$ is