Question

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

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

Answer 1

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

As $f\left(1\right)=-2$ a $f^{\prime }\left(x\right)\ge 2\forall x\in [1,6]$.

Applying Lagrange's mean value theorem
$\frac{f\left(6\right)-f\left(1\right)}{5}=f^{\prime }\left(c\right)\ge 2$
$=f\left(6\right)\ge 10+f\left(1\right)$
$=f\left(6\right)\ge 10-2$
$=f\left(6\right)\ge 8$.