Mathematics

Mathematics

Problem

A differentiable function gg satisfies g′(x)>0g'(x) > 0 on (−∞,−1)(-\infty,-1), g′(x)<0g'(x) < 0 on (−1,3)(-1,3), and g′(x)>0g'(x) > 0 on (3,∞)(3,\infty). Which interval describes where gg is decreasing? a) (−∞,−1)(-\infty,-1) b) (−1,3)(-1,3) c) (−1,3](-1,3] d) (−∞,3)(-\infty,3)
Intuitive explanation

The derivative g′(x)g'(x) tells you the slope of gg at each point—when it's positive, the function climbs; when it's negative, the function falls. You just need to identify which interval has g′(x)<0g'(x) < 0, since that's exactly where gg is moving downward. The sign of the derivative is your direct window into the function's behavior.

Intuitive explanation

The derivative g′(x)g'(x) tells you the slope of gg at each point—when it's positive, the function climbs; when it's negative, the function falls. You just need to identify which interval has g′(x)<0g'(x) < 0, since that's exactly where gg is moving downward. The sign of the derivative is your direct window into the function's behavior.

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