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Question

The number of integers satisfying the inequation x(x2+2x+2)(log2x3)(x+3)2(x2x6)0, is

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Solution

Clearly, x>0 (1)
and (x+2)(x3)0x2,3 (2)

x(x2+2x+2)(log2x3)(x+3)2(x2x6)0
x(log2x3)(x+2)(x3)0
(x2+2x+2>0,(x+3)2>0, x>0)

Critical points are
x=0
log2x3x=8
(x+2)(x3)x=2,3

By wavy curve method,
x(2,0](3,8] (3)
(1)(2)(3), we get
x(3,8]
Integers are 4,5,6,7,8

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