Differentiate a factored cubic and distinguish a repeated stationary point

$y'=-[(x+b)^2+2(x+a)(x+b)]=-(x+b)[x+b+2(x+a)]=-(x+b)(3x+2a+b)$. Thus stationary inputs are $x=-b$ or $x=-(2a+b)/3$.

At x=−b,y=0. At $x=-(2a+b)/3$, the source substitution gives $y=-[-(2a+b)/3+a][-(2a+b)/3+b]^2=-[(a-b)/3][(-2a+2b)/3]^2=4(b-a)^3/27$. Thus the two coordinates are $(-b,0)$ and $(-(2a+b)/3,4(b-a)^3/27)$.

The source calls both formulae turning points. This is correct for a≠b. If a=b, they coincide and $y=-(x+a)^3$ has a stationary inflection at (−a,0), with no change of direction; it is not a turning point. Both strict regimes in(ii)have a≠b.

All source intercepts are (0,−ab²),(−a,0),(−b,0). The double zero x=−b touches the axis and is a turn for the stated regimes; x=−a is a crossing.

For a>0,b<0, −a<0<−b and −ab²<0. The second stationary coordinate has negative height and is a local minimum; (−b,0) is a local maximum. The y-intercept is below the axis. The source drawing places the other turn left of the y-axis, but the assumptions do not fix that: $-(2a+b)/3$ can have either sign. The example panel shows one allowed placement.

For a<b<0, 0<−b<−a and −ab²>0. The source sketch has a minimum at (−b,0) and a maximum at $(-(2a+b)/3,4(b-a)^3/27)$ between the two positive zeros, with a positive y-intercept. Both curves rise to the left and fall to the right.