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If f(x) is a polynomial function which satisfy the relation (f(x))² f(x)=(f(x))³ f'(x), f'(0)=f'(1)=f'(-1)=0,f(0)=4, f(±1)=3, then f(i) (where i=√7) is equal to

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pankaj gupta , 13 Years ago

Grade 12th Pass

1 Answers

Jitender Singh

Last Activity: 10 Years ago

Ans:

$f'(0) = f'(-1) = f'(1) = 0$

$\Rightarrow f'(x) = ax(x-1)(x+1), a = constant$

$\int df(x) = \int ax(x-1)(x+1)dx$

$f(x) = a(\frac{x^{4}}{4}-\frac{x^{2}}{2}) + c, c = constant$

$f(0) = a(\frac{0^{4}}{4}-\frac{0^{2}}{2}) + c = 4$

$\Rightarrow c = 4$

$f(\pm 1) = a(\frac{(\pm 1)^{4}}{4}-\frac{(\pm 1)^{2}}{2}) + 4 = 3$

$\frac{-a}{4} = -1$

$a = 4$

$f(x) = x^{4}-2x^{2}+4$

$f'(x) = 4x^{3}-4x$

$f''(x) = 12x^{2}-4$

$f''(\sqrt{7}) = 12(\sqrt{7})^{2}-4 = 80$

Thanks & Regards

Jitender Singh

IIT Delhi

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