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  • Let  <img alt="y(x)=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)\left(1+x^{16}\right)" src="https://entrancecorner.oncodecogs.com/gif.latex?y%28x%29%3D%281&plus;x%29%5

Let  y(x)=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)\left(1+x^{16}\right).Then y^{\prime}-y^{\prime \prime}at x=-1 is equal to :

Option: 1

976


Option: 2

944


Option: 3

464


Option: 4

496


Answers (1)

best_answer

f(x)=y=\frac{(1-x)(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)\left(1+x^{16}\right)}{(1-x)}
f(x)=y=\frac{\left(1-x^{32}\right)}{1-x} \Rightarrow f(-1)=0
(1-x) y=1-x^{32}

differentiate both side

(1-x) y^{\prime}+y(-1)=-32 x^{31} \quad x=-1 \Rightarrow y^{\prime}=16

differentiate both side

(1-x) y^{\prime}+y^{\prime}(-1)-y^{\prime}=-(32)(31) x 30
Put \: \mathrm{x}=-1
2 y^{\prime \prime}-2 y^{\prime}=-(32)(31)
y^{\prime \prime}-y^{\prime}=-(16)(31)
y^{\prime}-y^{\prime \prime}=496

Posted by

Ritika Harsh

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