中值定理辅助函数类型
辅助函数表格
| 中值等式 G(\xi) = 0 | 辅助函数 F(x) |
|---|---|
| \displaystyle\frac{f'(x)}{f(x)} = 0 | \ln f(x) |
| f'(\xi) + A\xi^k + B = 0 | \displaystyle f(x) + \frac{Ax^{k+1}}{k+1} + Bx |
| f(a)g'(\xi)-f'(\xi)g(a) - k = 0 | f(a)g(x) - f(x)g(a) - kx |
| \displaystyle\sum_{i=0}^{n-1}a_i(n-i)\xi^{n-1-i} = 0 | \displaystyle\sum_{i=0}^{n-1}a_i x^{n-i} |
| f'(\xi)g(\xi) + f(\xi)g'(\xi) = 0 | f(x)g(x) |
| f(\xi)g''(\xi) - f''(\xi)g(\xi) = 0 | f(x)g'(x) - f'(x)g(x) |
| \xi f'(\xi) + kf(\xi) = 0 | x^kf(x) |
| (\xi - 1)f'(\xi) + kf(\xi) = 0 | (x-1)^kf(x) |
| f'(\xi)g(1-\xi) - kf(\xi)g'(1-\xi) = 0 | g^k(1-x)f(x) |
| f'(\xi) + \lambda f(\xi) = 0 | e^{\lambda x}f(x) |
| f'(\xi) + g'(\xi)f(\xi) = 0 | e^{g(x)}f(x) |
| \xi f'(\xi) - kf(\xi) = 0 | \displaystyle\frac{f(x)}{x^k} |
| f'(\xi) - kf(\xi) = 0 | \displaystyle\frac{f(x)}{e^{kx}} |
| \displaystyle f(\xi) + \frac{x - b}{a}f'(\xi) = 0 | (x - b)^a f(x) |
| f'(\xi)g(\xi) - f(\xi)g'(\xi) = 0 | \displaystyle\frac{f(x)}{g(x)} |
| \displaystyle\frac{1-\xi^2}{(1+ \xi^2)^2} = 0 | \displaystyle\frac{x}{1+x^2} |
| f'(\xi) - f(\xi) + k\xi - k = 0 | \displaystyle\frac{f(x) - kx}{e^{x}} |
| f''(\xi) + f'(\xi) - k = 0 | e^x[f'(x) - k] |
| f'(\xi) + k[f(\xi)-\xi] -1 = 0 | e^{kx}[f(x) - x] |
| f''(\xi) - f(\xi) = 0 | e^x[f(x) - f'(x)] |
证明
\begin{aligned}
&\frac{f'(x)}{f(x)} = 0 \\
&\int \frac{f'(x)}{f(x)} dx \\
=& \int \frac{df(x)}{f(x)} \\
=& \ln f(x) \\
&F(x) = \ln f(x)
\end{aligned}
\begin{aligned}
&f'(x) + Ax^k + B =0\\
&\int f'(x) + Ax^k + B dx \\
&= f(x) + \frac{Ax^{k+1}}{k+1} + Bx \\
&F(x) = f(x) + \frac{Ax^{k+1}}{k+1} + Bx
\end{aligned}
\begin{aligned}
&f(a)g'(x) - f'(x)g(a) - k = 0 \\
&\int f(a)g'(x) - f'(x)g(a) - k dx \\&
= f(a)g(x) - f(x)g(a) -kx \\
&F(x) = f(a)g(x) - f(x)g(a) -kx
\end{aligned}
\begin{aligned}
&\sum_{i=0}^{n-1}a_i(n-i)x^{n-1-i} = 0 \\
&\int \sum_{i=0}^{n-1}a_i(n-i)x^{n-1-i} dx = \sum_{i=0}a_ix^{n-i} \\
&F(x) = \sum_{i=0}a_ix^{n-i}
\end{aligned}
\begin{aligned}
f'(x)g(x) + f(x)g'(x) &= 0 \\
\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} &= 0 \\
[\ln f(x) + \ln g(x)]' &= 0 \\
[\ln f(x)\cdot g(x)]' &= 0 \\
F(x) = f(x)\cdot g(x) \\
\end{aligned}
\begin{aligned}
f(x)g''(x) - f''(x)g(x) &= 0 \\
f(x)g''(x) - f'(x)g'(x) \\
+ f'(x)g'(x) - f''(x)g(x) &= 0 \\
[f(x)g''(x) + f'(x)g'(x)] \\
- [f'(x)g'(x) + f''(x)g(x)] &= 0 \\
[f(x)g'(x)]' - [f'(x)g(x)]' &= 0 \\
[f(x)g'(x) - f'(x)g(x)]' &= 0 \\
F(x) = f(x)g'(x) - f'(x)g(x) \\
\end{aligned}
\begin{aligned}
xf'(x) + kf(x) &= 0 \\
\frac{f'(x)}{f(x)} + \frac{k}{x} &= 0 \\
[\ln f(x) + k\ln x]' &= 0 \\
[\ln f(x) + \ln x^k]' &= 0 \\
[\ln f(x)x^k]' &= 0 \\
F(x) = x^k f(x) \\
\end{aligned}
\begin{aligned}
(x-1)f'(x) + kf(x) &= 0 \\
\frac{f'(x)}{f(x)} + \frac{k}{x-1} &= 0 \\
[\ln f(x) + k\ln (x-1)]' &= 0 \\
[\ln f(x) + \ln (x-1)^k]' &= 0 \\
[\ln f(x)(x-1)^k]' &= 0 \\
F(x) = (x-1)^k f(x) \\
\end{aligned}
\begin{aligned}
f'(x)g(1-x) + kf(x)g'(1-x) &= 0 \\
\frac{f'(x)}{f(x)} + k\frac{g'(1-x)}{g(1-x)} &= 0 \\
[\ln f(x) + k\ln g(1-x)]' &= 0 \\
[\ln f(x) + \ln g^k(1-x)]' &= 0 \\
[\ln f(x)g^k(1-x)]' &= 0 \\
F(x) = g^k(1-x) f(x) \\
\end{aligned}
\begin{aligned}
f'(x) + \lambda f(x) &= 0 \\
\frac{f'(x)}{f(x)} + \lambda &= 0 \\
[\ln f(x) + \lambda x]' &= 0 \\
[\ln f(x) + \ln e^{\lambda x}]' &= 0 \\
[\ln e^{\lambda x} f(x)]' &= 0 \\
F(x) = e^{\lambda x} f(x)
\end{aligned}
\begin{aligned}
f'(x) + g'(x)f(x) &= 0 \\
\frac{f'(x)}{f(x)} + g'(x) &= 0 \\
[\ln f(x) + g(x)]' &= 0 \\
[\ln f(x) + \ln e^{g(x)}]' &= 0 \\
[\ln e^{g(x)}f(x)]' &= 0 \\
F(x) = e^{g(x)}f(x) \\
\end{aligned}
\begin{aligned}
xf'(x) - kf(x) &= 0 \\
\frac{f'(x)}{f(x)} - \frac{k}{x} &= 0 \\
[\ln f(x) - k\ln x]' &= 0 \\
[\ln \frac{f(x)}{x^k}]' &= 0 \\
F(x) = \frac{f(x)}{x^k}
\end{aligned}
\begin{aligned}
f'(x) - kf(x) &= 0 \\
\frac{f'(x)}{f(x)} - k &= 0 \\
[\ln f(x) - kx]' &= 0 \\
[\ln \frac{f(x)}{e^{kx}}]' &= 0 \\
F(x) = \frac{f(x)}{e^{kx}}
\end{aligned}
\begin{aligned}
f(x) + \frac{x-b}{a}f'(x) &= 0 \\
\frac{a}{x-b} + \frac{f'(x)}{f(x)} &= 0 \\
[a\ln(x-b) + \ln f(x)]' &= 0 \\
[\ln (x-b)^af(x)]' &=0\\
F(x) = (x-b)^af(x)
\end{aligned}
\begin{aligned}
f'(x)g(x) - f(x)g'(x) &= 0 \\
\frac{f'(x)}{f(x)} -\frac{g'(x)}{g(x)} &= 0 \\
[\ln f(x) - \ln g(x)]' &= 0 \\
[\ln \frac{f(x)}{g(x)}]' &= 0 \\
F(x) = \frac{f(x)}{g(x)}
\end{aligned}
\begin{aligned}
&\frac{1-x^2}{(1+x^2)^2} = 0 \\
& \int \frac{1-x^2}{(1+x^2)^2} dx \\
\xlongequal{x=\tan u}&\int \frac{1 - \tan^2 u}{\sec ^ 2 u} du \\
=&\int \cos^2 u - \sin^2 u \ du \\
=&\int \frac{1 + \cos 2u}{2} - \frac{1 - \cos 2u}{2} \ du \\
=&\int \cos 2u \ du \\
=& \frac{1}{2} \int \cos 2u \ d2u \\
=& \frac{1}{2} \sin 2u \\
=& \cos u \cdot \sin u \\
=& \frac{1}{\sqrt{1 + x^2}} \cdot \frac{x}{\sqrt{1 + x^2}} \\
=& \frac{x}{1 + x^2} \\
& F(x) = \frac{x}{1 + x^2}
\end{aligned}
\begin{aligned}
f'(x) - f(x) + kx - k &= 0 \\
[f'(x) - k] - [f(x) - kx] &= 0 \\
g'(x) - g(x) \xlongequal{g(x)=f(x) - kx} 0 \\
\frac{g'(x)}{g(x)} - 1 &= 0 \\
[\ln g(x) - x]' &= 0 \\
[\ln g(x) - \ln e^x]' &= 0 \\
[\ln \frac{g(x)}{e^x}]' &= 0 \\
F(x) = \frac{g(x)}{e^x} = \frac{f(x) - kx}{e^x} \\
\end{aligned}
\begin{aligned}
f''(x) + f'(x) - k &= 0 \\
[f''(x) - 0] + [f'(x) - k] & = 0 \\
g'(x) + g(x) \xlongequal{g(x)=f'(x) - k} 0 \\
\frac{g'(x)}{g(x)} + 1 &= 0 \\
[\ln g(x) + x]' &= 0 \\
[\ln g(x) + \ln e^x]' &= 0 \\
[\ln g(x)e^x]' &= 0 \\
F(x) = g(x)e^x = e^x[f'(x) - k] \\
\end{aligned}
\begin{aligned}
f'(x) + k[f(x) - x] - 1 &= 0 \\
[f'(x) - 1] + k[f(x) - x] &= 0 \\
g'(x) + kg(x) \xlongequal{g(x)=f(x) - x} 0 \\
\frac{g'(x)}{g(x)} + k &= 0 \\
[\ln g(x) + kx]' &= 0 \\
[\ln g(x) + \ln e^{kx}]' &= 0 \\
[\ln g(x)e^{kx}]' &= 0 \\
F(x) = g(x)e^{kx} = e^{kx}[f(x) - x] \\
\end{aligned}
\begin{aligned}
f''(x) - f(x) &= 0 \\
[f''(x) - f'(x)] + [f'(x) - f(x)] & = 0 \\
g'(x) + g(x) \xlongequal{g(x)=f'(x) - f(x)} 0 \\
\frac{g'(x)}{g(x)} + 1 &= 0 \\
[\ln g(x) + x]' &= 0 \\
[\ln g(x) + \ln e^x]' &= 0 \\
[\ln g(x)e^x]' &= 0 \\
F(x) = g(x)e^x = e^x[f'(x) - f(x)] \\
\end{aligned}
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