Let f(x) be a polynomial of degree 5 having extrema at x = 1 and x = −1.

Given:

lim_x→∞ [f(x)/x³] = −5

Find:

f(2) − f(−2)

Since extrema occur at x = ±1:

f'(x) contains factor:

(x²−1)

Degree of polynomial is 5, therefore:

f'(x) = −25(x²−1)²

Integrating:

f(x) = −5x⁵ + (50/3)x³ − 25x + C

Now:

f(2) − f(−2)

= −112

Final Answer

−112