Question

Suppose that on the interval $[-2,4]$ the function $f$ is differentiable, $f\left(2\right)=1$ and $|f\left(x\right)|$ $\le$ $5$. Find the bounding function of $f$ on $[-2,4],$ using LMVT.

• A. $y=-5x-9$ and $y=5x+11$
• B. $y=-5x+9$ and $y=5x+11$
• C. $y=5x-9$ and $y=5x-11$
• D. $y=5x+9$ and $y=5x-11$

Answer 1

The correct option is A $y=-5x-9$ and $y=5x+11$

Let $xϵ[-2,4]$ Consider the interval $[-2,x]$ By LMVT

$\frac{f\left(x\right)-f(-2)}{x-(-2)}=f\left(c\right),c\in (-2,4)$

$\Rightarrow -5\le \frac{f\left(x\right)-1}{x+2}\le 5\{\therefore |f(x)|\le 5\}$

$\therefore -5x-10\le f\left(x\right)-1\le 5x+10$

$-5x-9\le f\left(x\right)\le 5x+11$.