Mathematics - Application of Derivative Question with Solution | TestHub

Let $u\in (a,c),v\in (c,b)$ then by LMVT on $(\mathrm{a},\mathrm{c}),(\mathrm{c},\mathrm{b})$ it follows $f^1(u)=\frac{f(c)-f(a)}{c-a},f^1(v)=\frac{f(b)-f(c)}{b-c}.$ But $\mathrm{u}<\mathrm{v}$ and $f^{11}(x)<0$ for all $x\in (a,b)\Rightarrow f^1(x)\downarrow \Rightarrow f^1(u)>f^1(v)\Rightarrow \frac{f(c)-f(a)}{f(b)-f(c)}>\frac{c-a}{b-c}$.