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MathCraft: PDF to LaTeX file: Sample-02

17 Hits, Last modified: 5 months ago

rightarrow \mathbb{R}$ is convex, then $$ f\left(\frac{a+b}{2}\right) \leqslant \frac{1}{b-a} \int_{a}^{b} f(x) d x \leqslant \frac{f(a)+f(b)}{2} . $$ \noindent\textbf{Proof}: First of all,... rapfigure} \vspace{.2 in} Let now $r(x)=f\left(\frac{a+b}{2}\right)+c\left(x-\frac{a+b}{2}\right)$ be

MathCraft: PDF to LaTeX file: Sample-01

16 Hits, Last modified: 5 months ago

ns $$ \begin{aligned} & E(x, y ; r, s)=\left\{\dfrac{r\left(y^{s}-x^{s}\right)}{s\left(y^{r}-x^{r}\right)}\right\}^{\dfrac{1}{s-r}} \\ & E(x, y ; r, 0)=E(0, r)=\left\{\dfrac{y^{r}-x^{r}}{r(\ln y-\ln x)}\right\}^{1 / r} \\ & E(x, y ; r, r)=e^{-\dfrac{1}{r}}\left(\dfrac{x^{x^{r}}}{y^{y^{r}}}\right)^{