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Q.
Let $f\left(1\right)=-2$and $f'\left(x\right)\ge4.2$ for $1\le x\le6$The smallest possible value of $f\left(6\right)$ is
Application of Derivatives
Solution:
Using Lagrange’s mean value theorem, for some $c\in\left(1,6\right)$
$f^{'}\left(c\right)=\frac{f\left(6\right)-f\left(1\right)}{5}=\frac{f\left(6\right)+2}{5}\ge4.2$
or $f\left(6\right)+2\ge21$
or $f\left(6\right)\ge19$