Question

Let $f\text{'}\left(x\right)$ be differentiable for all$x$. If $f\left(1\right)=-2$ and$f\text{'}\left(x\right)\ge 2\forall x\in \left[1,6\right]$, then

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

Answer 1

The correct option is B

$f\left(6\right)\ge 8$

Explanation for the correct option:

$f\left(1\right)=-2$ and$f\text{'}\left(x\right)\ge 2\forall x\in \left[1,6\right]$

$f\left(x\right)$ be continuous on $\left[1,6\right]$and differentiable on $\left(1,6\right)$ , therefore, by Lagrange's mean value theorem:

$\exists c\in \left(1,6\right)$ such that $f\text{'}\left(c\right)=\frac{f\left(6\right)-f\left(1\right)}{6-1}$

$$\begin{gathered} f\left(6\right)-f\left(1\right)=5f\text{'}\left(c\right) \\ \ge 10 \end{gathered}$$

$$\begin{gathered} f\left(6\right)+2\ge 10 \\ f\left(6\right)\ge 8 \end{gathered}$$

Hence, option (B) is the correct answer