Question

Let $f\left(1\right)=-2$and $f^{\prime }\left(x\right)\ge 4.2$ for $1\le x\le 6$The smallest possible value of $f\left(6\right)$ is

• A. $9$
• B. $12$
• C. $15$
• D. $19$

Answer 1

Answer: (D)

Using Lagrange’s mean value theorem, for some $c\in \left(1,6\right)$
$f^{{}^{\prime }}\left(c\right)=\frac{f\left(6\right)-f\left(1\right)}{5}=\frac{f\left(6\right)+2}{5}\ge 4.2$
or $f\left(6\right)+2\ge 21$
or $f\left(6\right)\ge 19$