Question

If $f$ be the differentiable for $\forall x$. If $f\left(1\right)=-2$ and $f^{\prime }\left(x\right)\ge 2$ for $[1,6]$ then

• A. $f\left(6\right)<8$
• B. $f\left(6\right)=5$
• C. $f\left(6\right)\ge 8$
• D. $f\left(6\right)<5$`

Answer 1

The correct option is C $f\left(6\right)\ge 8$

$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}{f}^{\prime }\left(c\right)=\frac{f\left(6\right)-f\left(1\right)}{6-1}\frac{f\left(b\right)+2}{5}\ge 2f\left(6\right)\ge 8$