Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x^(6) is unity and it has extrema at x = -1 and x = 1. If mathop lim limits_x -> 0 f(x) / x^3 = 1, then 5.f(2) is equal to _________.

f(x) = x^6 + ax^5 + bx^4 + x^3Therefore f'(x) = 6x^5 + 5ax^4 + 4bx^3 + 3x^2Roots 1 -1 Therefore 6 + 5z + 4b + 3 = 0 - 6 + 5a - 4b + 3 = 0 solvinga = - 3 / 5b = - 3 / 2 Therefore f(x) = x^6 - 3 / 5x^5 - 3 / 2x^4 + x^3 Therefore 5.f(2) = 5[ 64 - 96 / 5 - 24 + 8 ] = 144

Let fx be a polynomial of degree 6 in x in which the co... | null (Mathematics)

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 &

-

\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

Explanation

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