作业帮15.若0
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15.若0
15.若0
15.若0
Let y = ƒ(x) = (e^x)/x²
dy/dx = [(e^x)x² - 2x(e^x)]/x⁴
= (e^x)(x - 2)/x³
Let dy/dx = 0,x = 2
When x < 2,dy/dx < 0
So,y = ƒ(x) = (e^x)/x² is strictly decreasing function.
[y = ƒ(x) = (e^x)/x² 在(0,2)上,是一个严格递减函数]
∴ when 0 < x₁< x₂< 2,ƒ(x₁) > ƒ(x₂)
ie.e^x₁/x²₁> e^x₂/x²₂
附：事实上
d²y/dx² = {[(e^x)(x - 2)+ (e^x)]x³ - 3(e^x)(x - 2)x²}/(x³)²
= {[(e^x)(x - 2)+ (e^x)]x - 3(e^x)(x - 2)}/x⁴
= (e^x){[(x - 2)+ 1]x - 3(x - 2)}/x⁴
= (e^x)[(x - 1)x - 3(x - 2)]/x⁴
= (e^x)(x² - 4x + 6)/x⁴> 0
So,y = ƒ(x) = (e^x)/x² is a concaving-up function.
[y = ƒ(x) = (e^x)/x² 是一个上凹函数]
手机上看不见图