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Correct answer: a is in (257, infinity)
f(x) = [(x⁴+1)/a]*sin(x) + cos(x) + cosh(x). cos(x) and cosh(x) are even. For f to be even, [(x⁴+1)/a]*sin(x) must be even. Since sin(x) is odd, the GIF term must be an even function. But (x⁴+1)/a is always positive (for a>0), so [(x⁴+1)/a] >= 0. For the product to be even: [(x⁴+1)/a] must be an even function of x (i.e., its value at x equals its value at -x, which it is since x⁴+1 is even). The simplest sufficient condition making the product even is [(x⁴+1)/a] = constant, specifically 0 everywhere on [-4,4]. The maximum value of x⁴+1 on [-4,4] is at x = ±4: 256+1 = 257. For [(x⁴+1)/a] = 0 for all x in [-4,4], we need (x⁴+1)/a < 1 for all x, i.e., a > max(x⁴+1) = 257. So a >= 257 (with a = 257 giving max value 1, floor = 1, not 0). Thus a > 257, i.e., a in (257, infinity). But if a = 257, at x = ±4, floor(257/257) = floor(1) = 1, so the term is 1*sin(x), making f odd at those points. For a = 257: [(x⁴+1)/257] is 0 for |x|<4 (since max interior value <257) and 1 at endpoints. Actually at x=4: (256+1)/257 = 1, floor = 1. So the GIF = 1 at x = ±4 and 0 elsewhere. Then f is even iff the product is even. [(x⁴+1)/a] is even in x (same value at x and -x), and sin(x) is odd, so their product is odd, not even (unless GIF = 0 everywhere). At a = 257, GIF = 1 at ±4, product = sin(x) at those points (odd), so f is NOT even. For f to be even we need a > 257 strictly, a in (257, infinity).