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Let f : [1, ) [2, ) be a differentiable function. If 10 1x f(t) dt = 5xf(x) - x5 - 9 for all x 1, then the value of f(3) is :
(3) 32
Put x = 1: 0 = 5f(1) - 1 - 9 f(1) = 2
Differentiate both sides w.r.t. x:
10f(x) = 5f(x) + 5xf'(x) - 5x4
xf'(x) - f(x) = x4
xf'(x) - f(x)x2 = x2
ddx ( f(x)x ) = x2
f(x)x = x33 + C
At x = 1: 2 = 13 + C C = 53
f(x) = x43 + 5x3
f(3) = 813 + 153 = 27 + 5 = 32
Answer: (3)
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