Mathematics

Mathematics

Problem

The table below gives selected values of f′(x)f'(x).
x−2−1012
f'(x)−3−1025
On which interval is ff most clearly increasing? a) (−1,2)(-1,2) b) (1,2)(1,2) c) (0,2)(0,2) d) (−2,−1)(-2,-1)
Intuitive explanation

The derivative f′(x)f'(x) tells you the slope of ff at each point—positive values mean ff is climbing, negative values mean it's falling. To find where ff is "most clearly increasing," look for where f′(x)f'(x) is not just positive, but noticeably large and positive, showing steep upward growth. Compare the positive derivative values across the table to identify the interval with the strongest increase.

Intuitive explanation

The derivative f′(x)f'(x) tells you the slope of ff at each point—positive values mean ff is climbing, negative values mean it's falling. To find where ff is "most clearly increasing," look for where f′(x)f'(x) is not just positive, but noticeably large and positive, showing steep upward growth. Compare the positive derivative values across the table to identify the interval with the strongest increase.

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