Question

If $f^{\prime \prime }\left(x\right)<0,\forall xϵ(a,b)$, and $(c,f(c\left)\right)$ is point of maxima, where $cϵ(a,b)$, then $f^{\prime }\left(c\right)$ is

• A. $\frac{f\left(b\right)-f\left(a\right)}{b-a}$
• B. $\left[\frac{f\left(b\right)-f\left(a\right)}{b-a}\right]$
• C. $2\left[\frac{f\left(b\right)-f\left(a\right)}{b-a}\right]$
• D. $0$

Answer 1

The correct option is A $\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$F\left(c\right)=\frac{c-a}{2}[f\left(a\right)-f\left(c\right)]+\frac{b-c}{2}[f\left(b\right)-f\left(c\right)]$

$f^{\prime \prime }\left(x\right)<0$ and $(c,f(c\left)\right)$ is point of maxima

$F^{\prime }\left(c\right)=(b-a)f^{\prime }\left(c\right)+f\left(a\right)+f\left(b\right)⟶1$

$F^{\prime \prime }\left(c\right)=f^{\prime \prime }\left(c\right)(b-a)<0$

$\Rightarrow F^{\prime }\left(c\right)=0$

From equation $1\Rightarrow f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ (Answer)