Calculus / Unit 6: Applications of Derivatives II
6.2 The First Derivative Test
Notes
Local extrema occur at critical numbers where $f'$ changes sign.
- $f'$ changes from $+$ to $-$: local maximum.
- $f'$ changes from $-$ to $+$: local minimum.
- No sign change: no local extremum there.
- Example: $f(x) = x^3 - 3x$ has $f' = 3(x - 1)(x + 1)$: signs $+, -, +$. So a local max at $x = -1$ (value $f(-1) = 2$) and a local min at $x = 1$ (value $f(1) = -2$).
Practice
At a critical number, $f'$ changes from positive to negative. What is there?
The function rises then falls, so it peaks.
For $f(x) = x^3 - 3x$, what is the local maximum value?
The local max is at $x = -1$, and $f(-1) = -1 + 3 = 2$.
Let $f'(x) = (x - 1)^2(x + 2)$. Which statement is true?
At $-2$, $f'$ goes from $-$ to $+$. At 1, $(x - 1)^2$ never changes sign, so $f'$ stays positive.
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