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MathCraft: PDF to LaTeX file: Sample-01

4 Hits, Last modified: 5 months ago

} Let us consider the following means $$ \begin{aligned} & E(x, y ; r, s)=\left\{\dfrac{r\left(y^{s}-x^... -y^{r}\right)} \\ & E(x, y ; 0,0)=\sqrt{x y} \end{aligned} $$ where $x$ and $y$ are positive real number... q 0 \\ \ln x, & r=0\end{cases} $$ Now $$ \begin{aligned} f^{\prime}(x) & =p^{2} x^{r-1}+2 p q x^{t-1}+q... (r-1) / 2}+q x^{(s-1) / 2}\right)^{2} \geq 0 \end{aligned} $$ This implies $f$ is monotonically increasi

MathCraft: PDF to LaTeX file: Sample-02

4 Hits, Last modified: 5 months ago

s(x) d x . \tag{1} \end{equation*} Now $$ \begin{aligned} \int_{a}^{b} r(x) d x & =\int_{a}^{b}\left[f\l... d x \\ & =f\left(\frac{a+b}{2}\right)(b-a), \end{aligned} $$ and $$ \begin{aligned} \int_{a}^{b} s(x) d x & =\int_{a}^{b}\left[f(a)+\frac{f(b)-f(a)}{b-a}(x-a)\... b}(x-a) d x \\ & =\frac{f(a)+f(b)}{2}(b-a) . \end{aligned} $$ Using above value in (1), we have $$ f\left