Question

How do you find all critical point and determine the min, max and inflection given `D(r)=-r^2-2r+8`?

Answer 1

Critical points `r_j` make the derivative `D'(r_j)=0`. We them have to determine whether they are maxima, minima or inflexion.

Explanation:

The first derivative is `D'(r)= -2r-2`, so we find the critical points by solving `-2r-2=0`, which only solution is `r=-1`. We then calculate the second derivative in `r=-1`; the second derivative is `D''=-2`, which is negative everywhere, so it is negative in the critical point `r=-1`. The function therefore has a maximum at the point `r=-1`.

Please observe that this is coherent, as the graph is a parabola pointing downwards, so it has a single maximum