Practice question · Sort into groups
A function has derivative f'(x) = (x - 1)(x - 3). Sort each point by whether f is increasing or decreasing there.
Groups: f is increasing there · f is decreasing there
- x = 2.5
- x = -3
- x = 1.5
- x = 0
- x = 2
- x = 4
Hints
- You only need the SIGN of the product, so check the sign of each bracket.
- Two negative brackets multiply to a positive.
Show the answer
f is increasing there: x = 0, x = 4, x = -3
f is decreasing there: x = 2, x = 2.5, x = 1.5
Why
Outside the interval from 1 to 3 both brackets share a sign, so the product is positive and f rises; between 1 and 3 the brackets differ in sign and f falls. So f' goes positive-negative-positive: a local maximum at x = 1 and a local minimum at x = 3, read off without ever computing f itself.
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