What does "opposite semi-planes of segment" mean?
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To determine the 'intervals of increase' and 'intervals of decrease' we can refer to the graph with respect to the x - axis.
• Knowing that t = x - axis, the 'intervals of increase' as an inequality would be 1 < x < 3, and 4 < x < ∞. Therefore we have our intervals of increase as (1,3) and (4, ∞).
• Respectively our 'intervals of decrease' as inequalities would be - ∞ < x < 1, and 3 < x < 4. Our intervals of decrease would then be (- ∞, 1) and (3,4).
• We are left with our local extrema and absolute extrema. Now remember the absolute extrema is the absolute lowest point in the whole graph, while the local extrema is the lowest point in a restricted interval. In this case our local extrema is our maximum, (3,1). But this maximum is not greater than the starting point (0, 4) so it appears, and hence their is no absolute extrema.
