“积分符号内取微分”的反例?

hbghlyj

设$f(x,y)=y^3\mathrm e^{-y^2x},F(y)=\int_0^∞f(x,y)\rmd x$.
$$F(y)=y^3\int_0^∞\mathrm e^{-y^2x}\rmd x=\left.y^3⋅\frac{\mathrm e^{-y^2x}}{-y^2}\right|_{x=0}^∞=y^3⋅\frac1{y^2}=y⇒F'(0)=1$$
然而$$\frac{∂f}{∂y}(x,0)=\left.\mathrm e^{x y^2}(3 y^2 - 2 x y^4)\right|_{y=0}=0⇒\int_0^∞\frac{∂f}{∂y}(x,0)\rmd x=0$$

hbghlyj

设$f:[0,∞)\times[0,∞)\to[0,∞)$连续可导, $F(y)=\int_0^∞f(x,y)\rmd x$存在, 是否成立
\[F'(y)\geqslant\int_0^∞\frac{∂f}{∂y}(x,0)\rmd x\]似乎由Fatou's lemma可证得