The derivative f′ of a twice–differentiable function f is described as follows.
• f′(x)<0 for x<−2.
• f′(x)=0 at x=−2.
• f′(x)>0 for −2<x<1.
• f′(x)=0 at x=1.
• f′(x)<0 for 1<x<4.
• f′(x)=0 at x=4.
• f′(x)>0 for x>4.
The domain of f is all real numbers, and f is continuous and differentiable everywhere.
(a) Identify the intervals on which f is increasing and the intervals on which f is decreasing. Provide each interval in interval‐notation form and justify your conclusions.
(b) Determine the x–coordinates of all local extrema of f, classifying each as a local maximum or a local minimum. Support each classification with appropriate reasoning.
(c) Based only on the given information, describe the possible end behavior of f as x→−∞ and as x→∞. Explain how the long‐run sign pattern of f′ informs your description.