Question

比较3的兀次方与兀的三次方大小

Answer 1

解：
令f(x)=x^3-3^x

显然f(x)的一个零点为3
f'(x)=3x^2-3^x*ln3
f'(3)=27-27ln3<0
f''(x)=6x-3^x*(ln3)^2
f'''(x)=6-3^x*(ln3)^3
显然f'''(x)在x>3时小于零,
则f''(x)在x>3时单调递减,
而f''(3)=18-27(ln3)^2<0,
则x>3时f''(x)小于零,
则f'(x)在x>3时单调递减,
而f'(3)<0,
则f(x)在x>3时单调递减,
所以π^3<3^π

Answer 2

∵π>3
∴lgπ>lg3
lgπ/lg3>1
(lgπ/lg3)÷(lg3/lgπ)
=(lgπ/lg3)×(lgπ/lg3)
=(lgπ/lg3)²>1
∴lgπ/lg3>lg3/lgπ
即3^π>π^3