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f(x)−f(y) ≥ ln x−ln y+x−y \mathrm{f}(\mathrm{x})-\mathrm{f}(\mathrm{y}) \geq \ln \mathrm{x}-\ln \mathrm{y}+\mathrm{x}-\mathrm{y} \\ f(x)−f(y) ≥ ln x−ln y+x−y
f(x)−f(y)x−y ≥ ln x−ln yx−y+1 \frac{\mathrm{f}(\mathrm{x})-\mathrm{f}(\mathrm{y})}{\mathrm{x}-\mathrm{y}} \geq \frac{\ln \mathrm{x}-\ln \mathrm{y}}{\mathrm{x}-\mathrm{y}}+1\\ x−yf(x)−f(y) ≥ x−yln x−ln y+1
Let x>y\mathrm{x}>\mathrm{y}\\x>y
lim y → x f′(x−) ≥ 1x+1\lim _{y \rightarrow \mathrm{x}} f^{\prime}\left(x^{-}\right) \geq \frac{1}{x}+1\\lim y → x f′(x−) ≥ x1+1
Let x<y\mathrm{x}<\mathrm{y}\\x<y
lim y → x f′(x+) ≤ 1x+1 \lim _{y \rightarrow x} f^{\prime}\left(x^{+}\right) \leq \frac{1}{x}+1 \\ lim y → x f′(x+) ≤ x1+1
f1(x−)=f1(x+) f^1\left(x^{-}\right)=f^1\left(x^{+}\right) \\ f1(x−)=f1(x+)
f1(x)=1x+1 f^1(x)=\frac{1}{x}+1 \\ f1(x)=x1+1
f′(1x2)=x2+1 f^{\prime}\left(\frac{1}{x^2}\right)=x^2+1 \\ f′(x21)=x2+1
∑x=120(x2+1)=∑x−120 x2+20 \sum_{x=1}^{20}\left(x^2+1\right)=\sum_{x-1}^{20} x^2+20 \\ ∑x=120(x2+1)=∑x−120 x2+20
=20 × 21 × 416+20 =\frac{20 \times 21 \times 41}{6}+20 \\ =620 × 21 × 41+20
=2890
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