A twice-differentiable function f has a second derivative f′′ whose sign changes as follows:
• f′′(x)>0 for x<−3.
• f′′(−3)=0.
• f′′(x)<0 for −3<x<1.
• f′′(1)=0.
• f′′(x)<0 for 1<x<4.
• f′′(4)=0.
• f′′(x)>0 for x>4.
No other zeros of f′′ exist. Assume f and f′ are continuous everywhere.
(a) Determine the intervals on which the graph of f is concave up and the intervals on which it is concave down. Explain your reasoning.
(b) Identify all x-values at which the graph of f has a point of inflection. Justify each value you state.