The derivative is zero when x = 4, and the derivative is undefined at x = −3. (There’s an asymptote there,
so we can ignore the point. If the curve were defined at x = −3, then it would be a critical point, as you’ll
see in the next example.)
Now for the second derivative.
This is zero when x = . The second derivative is positive (and the graph is concave up) when x < ,
and it’s negative (and the graph is concave down) when x>.
We can now plug x = 4 into the second derivative. It’s positive there, so (4, 0) is a minimum.
Your graph should look like the following:
PROBLEM 4. Sketch the graph of y = (x-4). Plot all extrema, points of inflection, and asymptotes.