当x→0时,F(x)=∫0→x (x^2-t^2)f"(t)dt的导数F'(x)与x^2为等价无穷小,则f"(0)=
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F(x) = ∫(0→x) (x² - t²)ƒ''(t) dt = x²∫(0→x) ƒ''(t) dt - ∫(0→x) t²ƒ''(t) dt
F'(x) = 2x∫(0→x) ƒ''(t) + x²ƒ(x) - x²ƒ(x) = 2x∫(0→x) ƒ''(t) dt
lim(x→0) F'(x)/x² = 1，由于等价无穷小，极限结果是1
= lim(x→0) [2x∫(0→x) ƒ''(t) dt]/x² = 1
= lim(x→0) [2∫(0→x) ƒ''(t)]/x = 1
= lim(x→0) 2ƒ''(x) = 1
2ƒ''(0) = 1
ƒ''(0) = 1/2
F'(x) = 2x∫(0→x) ƒ''(t) + x²ƒ(x) - x²ƒ(x) = 2x∫(0→x) ƒ''(t) dt
lim(x→0) F'(x)/x² = 1，由于等价无穷小，极限结果是1
= lim(x→0) [2x∫(0→x) ƒ''(t) dt]/x² = 1
= lim(x→0) [2∫(0→x) ƒ''(t)]/x = 1
= lim(x→0) 2ƒ''(x) = 1
2ƒ''(0) = 1
ƒ''(0) = 1/2
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哈哈 谢谢。十分同意
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