Câu hỏi

Cho 0<a<b. Chứng minh rằng a a − b < ln b a < b a − b

Cho 0<a<b. Chứng minh rằng $\frac{a - b}{a} < ln \frac{a}{b} < \frac{a - b}{b}$

R. Robo.Ctvx9

Giáo viên

Xác nhận câu trả lời

Giải thích

∀ x ∈ ( a , b ) Ta có bất đẳng thức \frac1x>\frac1b\Rightarrow\frac1a\int_a^bdx>\int_a^b\frac{dx}x>\frac1b\int_a^bdx\\\Leftrightarrow\frac{b-a}a>\ln\left|x\right|\mbox{\large$\vert$}\nolimits_a^b \ln\frac ba

$\forall x \in (a, b)$ Ta có bất đẳng thức

$1 over a greater than 1 over x greater than 1 over b rightwards double arrow 1 over a integral subscript a superscript b d x greater than integral subscript a superscript b fraction numerator d x over denominator x end fraction greater than 1 over b integral subscript a superscript b d x left right double arrow fraction numerator b minus a over denominator a end fraction greater than ln open vertical bar x close vertical bar large vertical line subscript a superscript b less than fraction numerator b minus a over denominator b end fraction left right double arrow fraction numerator b minus a over denominator a end fraction greater than ln b over a less than fraction numerator b minus a over denominator b end fraction left right double arrow fraction numerator a minus b over denominator a end fraction less than ln a over b less than fraction numerator a minus b over denominator b end fraction$