I have the following question here...
Sketch the graph of a function f that has the following given properties. Identify critical numbers, local extrema, in ection points, and asymptotes. Assume $f$ and its derivatives exist and are continuous everywhere unless the contrary is implied or explicitly stated.
a) $f(0)=0, f(2)=3, f(4)$ is undefined, $f(7)=0, f(10)=1.$
b) $\displaystyle \lim_{x \to 4^-} f(x) = -\infty$ and $\displaystyle \lim_{x \to 4^+} f(x) = \infty.$
c) $\displaystyle \lim_{x \to -\infty} (f(x)+x+2)=0$, $\displaystyle \lim_{x \to \infty} (f(x)-2)=0.$
d) $f'(x)>0$ on $(0,2)$ and $(7,\infty)$; $f'(x)<0$ on $(-\infty,0),(2,4)$ and $(4,7)$.
e) $f'(0)$ is undefined and $\displaystyle \lim_{x \to 0^-} f'(x)=0$ and $\displaystyle \lim_{x \to 0^+} f'(x)=\infty.$
f) $\displaystyle f''(x)>0$ on $(-\infty,0)$ and $(4,10)$; $f''(x)<0$ on $(0,4)$ and $(10,\infty).$
So I summarized the following information...
$f(x)$ is increasing from $(0,2),(7,\infty).$
$f(x)$ is decreasing from $(-\infty,0),(2,4),(4,7).$
There is a vertical tangent at $x=0$.
$f(x)$ is concave up from $(-\infty,0),(4,10).$
$f(x)$ is concave down from $(0,4),(10,\infty).$
There are no points of inflection.
There is a minimum at $x=0$ and $x=7$.
The is a maximum at $x=2$.
There is a slant asymptote of $y=-x-2$ and $y=2$.
There is a vertical asymptote of $x=4.$
I got a graph that looks something like this. Can someone verify if this is correct?
Thank you!