Define two functions on (0, infinity): f1(x) = integral from 0 to x of [product from j=1 to 21 of (t - j)] dt, x 0 f2(x) = 98*(x - 1)⁵⁰ - 600*(x - 1)⁴⁹ + 2450, x 0 Let m1 = number of local minima of f1 in (0, infinity), n1 = number of local maxima of f1 in (0, infinity), m2 = number of

Question

Define two functions on (0, infinity): f1(x) = integral from 0 to x of [product from j=1 to 21 of (t - j)] dt, x > 0 f2(x) = 98*(x - 1)⁵⁰ - 600*(x - 1)⁴⁹ + 2450, x > 0 Let m1 = number of local minima of f1 in (0, infinity), n1 = number of local maxima of f1 in (0, infinity), m2 = number of local minima of f2 in (0, infinity), n2 = number of local maxima of f2 in (0, infinity). Find the value of 6*m1 + 4*n2 + 8*m2*n2.

Accepted Answer

48. f1'(x) = product_(j=1)²¹(x-j) changes sign at each integer from 1 to 21. On (0,inf) these are x=1,2,...,21 giving 20 sign changes: alternating min/max starting from a sign change at x=1. f1'(x) for x in (0,1) is product of (x-j) for j=1..21 — all factors negative, product is negative (odd number 21). So f1 is decreasing on (0,1), then at x=1 sign changes to positive (local min), then negative at x=2 (local max), alternating. This gives m1=11 local minima (at x=1,3,5,...,21) and n1=10 local maxima (at x=2,4,...,20) in (0,infinity). For f2: f2'(x)=(x-1)⁴⁸*[4900(x-1)-29400]=0 gives x=1 (even power, no sign change) and x=7 (sign change: negative to positive, so local min). Thus m2=1, n2=0. T

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