Definite Integration Question 44
Question 44
- If $f^{\prime \prime}(x)<0, \forall x \in(a, b)$, and $(c, f(c))$ is point of maxima, where $c \in(a, b)$, then $f^{\prime}(c)$ is (a) $\frac{f(b)-f(a)}{b-a}$ (b) $3 \frac{f(b)-f(a)}{b-a}$ (c) $2 \frac{f(b)-f(a)}{b-a}$ (d) 0
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Solution:
- $F^{\prime}(c)=(b-a) f^{\prime}(c)+f(a)-f(b)$
$$ \begin{aligned} F^{\prime \prime}(c) & =f^{\prime \prime}(c)(b-a)<0 \ \Rightarrow \quad F^{\prime}(c) & =0 \Rightarrow \quad f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} \end{aligned} $$
