Let g(x) = 300-8x^3-x^4
-Find the local maximum and minimum values.
-Find the intervals of concavity and the inflection points.
-Use this information to carefully sketch the graph of g.
g'(x)= -24x^2 - 4x^3 = -4x^2 (6 + x)
g'(x)=0 => x=0 or x=-6 therefore (0,300) and (-6,732) are the local extrema.
g''(x)= -48x - 12x^2 = -12x (4 + x)
g''(x)=0 => x=0 or x=-4, therefore (0,300) and (-4,556) are inflection points.
g''(1)= -60, g''(-1)= 36, g''(-6)=-144,
Therefore (0,oo) is ...
This shows how to find points of inflection, local maximum and minimum, and use these to sketch the graph.