"Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = $-$1 and x = 1. If $\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^3}}} = 1$, then $5.f(2)$ is equal to _________."

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Accepted Answer

"$f(x) = {x^6} + a{x^5} + b{x^4} + {x^3}$

$\therefore$ $f'(x) = 6{x^5} + 5a{x^4} + 4b{x^3} + 3{x^2}$

Roots 1 & $-$1

$\therefore$ $6 + 5z + 4b + 3 = 0$ & $- 6 + 5a - 4b + 3 = 0$ solving

$a = - {3 \over 5}$

$b = - {3 \over 2}$

$\therefore$ $f(x) = {x^6} - {3 \over 5}{x^5} - {3 \over 2}{x^4} + {x^3}$

$\therefore$ $5.f(2) = 5\left[ {64 - {{96} \over 5} - 24 + 8} \right] = 144$"

Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x⁶ is unity and it has extrema at x = $-$1 and x = 1. If $\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^3}}} = 1$, then $5.f(2)$ is equal to _________.

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Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = $-$1 and x = 1. If $\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^3}}} = 1$, then $5.f(2)$ is equal to _________.

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Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x⁶ is unity and it has extrema at x = $-$1 and x = 1. If $\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^3}}} = 1$, then $5.f(2)$ is equal to _________.