Question

Let the function, $f:[-7,0]\to R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }\left(x\right)\le 2,$ for all $x\in (-7,0),$ then for all such functions $f,f(-1)+f\left(0\right)$ lies in the interval:

• A. $[-6,20]$
• B. $(-\infty ,20]$
• C. $(-\infty ,11]$
• D. $[-3,11]$

Answer 1

The correct option is B $(-\infty ,20]$

$f(-7)=-3$ and $f^{\prime }\left(x\right)\le 2$
Applying LMVT in $[-7,0]$, we get
$\frac{f(-7)-f\left(0\right)}{-7}=f^{\prime }\left(c\right)\le 2\Rightarrow \frac{-3-f\left(0\right)}{-7}\le 2\Rightarrow f\left(0\right)+3\le 14\Rightarrow f\left(0\right)\le 11$

Applying LMVT in $[-7,-1]$, we get
$\frac{f(-7)-f(-1)}{-7+1}=f^{\prime }\left(c\right)\le 2\Rightarrow \frac{-3-f(-1)}{-6}\le 2\Rightarrow f(-1)+3\le 12\Rightarrow f(-1)\le 9$

Therefore, $f(-1)+f\left(0\right)\le 20$