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Let f : R R be a thrice differentiable odd function satisfying f'(x) 0, f''(x) = f(x), f(0) = 0, f'(0) = 3. Then 9f( e 3) is equal to .
36
f''(x) = f(x) f(x) = c1ex + c2e-x
f is odd f(-x) = -f(x) c2 = -c1
f(x) = c1(ex - e-x), which also gives f(0) = 0
f'(0) = 2c1 = 3 c1 = 32
f(x) = 32(ex - e-x) (and f'(x) = 32(ex + e-x) 0, as required)
f( e 3) = 32 (3 - 13 ) = 32 83 = 4
9f( e 3) = 36
Answer: 36
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