`f(x) = (1/2)x^4 + 2x^3` Find the points of inflection and discuss the concavity of the graph of the function.
Find the critical values for x by setting the second derivative of the function equal to zero and solving for the x value(s).
The critical value for the second derivative is x=0 and x=-2.
If f''(x)>0, the curve is concave up in the interval.
If f''(x)<0, the curve is concave down in the interval.
Choose a value for x that is less than -2.
f''(-3)=18 Since f''(-3)>0 the graph is concave up in the interval (-`oo,-2` ).
Choose a value for x that is between -2 and 0.
f''(-1)=-6 Since f''(-1)<0 the graph is concave down in the interval (-2, 0).
Choose a value for x that is greater than 0.
f''(1)=18 Since f''(1)>0 the graph is concave up in the interval (0, `oo).`
Because the direction of concavity changes twice and because f''(-2)=0 and
f"(0)=0 there will be an inflection point at x=-2 and x=0.
The inflection points are (-2, -8) and (0, 0).