Question

On the interval $I=[-2,2]$,for the function $\{\begin{array}{cc}(x+1)e^{-[\frac{1}{\left[x\right]}+\frac{1}{x}]}& (x\ne 0)\\ 0& (x=0)\end{array}$
( where $[$ $]$ is $GIF$ ) which one of the following hold good?

• A. is continuous for all values of $x\in I$
• B. is continuous for all values of $x\in I-\left(0\right)$
• C. assumes all intermediate values from $f(-2)$ & $f\left(2\right)$
• D. has a maximum value equal to $\frac{3}{e}$

Answer 1

Answer: (B), (C), (D)

is continuous for all values of $x\in I-\left(0\right)$
assumes all intermediate values from $f(-2)$ & $f\left(2\right)$
has a maximum value equal to $\frac{3}{e}$

$\underset{x\to 0^{+}}{lim}(x+1)e^{-\left|\frac{2}{x}\right|}=\underset{x\to 0^{+}}{lim}\frac{x+1}{e^{2/x}}=\frac{1}{e^{\infty }}=0$
$\underset{x\to 0^{-}}{lim}(x+1)e^{-(-\frac{1}{x}+\frac{1}{x})}=1$
Hence continuous for $xϵI-0$

$f(-2)=-e$ and $f\left(2\right)=3/e$

Therefore, has maximum value $3/e$