The first derivative test and the second derivative test are used to classify
critical points. A critical point is a point where the graph may have a local maximum,
local minimum, or neither.
1. Critical points
Critical points occur where:
\[
f'(c)=0
\]
or where \(f'(c)\) is undefined, as long as \(f(c)\) exists.
2. First derivative test
The first derivative tells whether the function is increasing or decreasing.
\[
f'(x)>0 \Rightarrow f(x)\text{ is increasing}
\]
\[
f'(x)<0 \Rightarrow f(x)\text{ is decreasing}
\]
Around a critical point \(c\):
| Sign change in \(f'(x)\) |
Graph behavior |
Classification |
| \(+\to-\) |
Increasing then decreasing |
Local maximum |
| \(-\to+\) |
Decreasing then increasing |
Local minimum |
| \(+\to+\) or \(-\to-\) |
No change from increasing to decreasing or reverse |
Neither |
3. Second derivative test
The second derivative measures concavity. At a critical point \(c\), where \(f'(c)=0\):
\[
f''(c)>0 \Rightarrow \text{local minimum}
\]
\[
f''(c)<0 \Rightarrow \text{local maximum}
\]
\[
f''(c)=0 \Rightarrow \text{test inconclusive}
\]
When the second derivative test is inconclusive, use the first derivative sign chart.
4. Example: \(f(x)=x^3-3x\)
Start with:
\[
f(x)=x^3-3x
\]
Differentiate:
\[
f'(x)=3x^2-3
\]
Find critical points:
\[
3x^2-3=0
\]
\[
x^2=1
\]
\[
x=-1,\qquad x=1
\]
Now use the first derivative sign chart.
| Interval |
Test value |
Sign of \(f'(x)\) |
Behavior |
| \((-\infty,-1)\) |
\(-2\) |
Positive |
Increasing |
| \((-1,1)\) |
\(0\) |
Negative |
Decreasing |
| \((1,\infty)\) |
\(2\) |
Positive |
Increasing |
At \(x=-1\), the sign changes from positive to negative:
\[
+\to-
\]
Therefore, \(x=-1\) gives a local maximum.
At \(x=1\), the sign changes from negative to positive:
\[
-\to+
\]
Therefore, \(x=1\) gives a local minimum.
Compute the function values:
\[
f(-1)=(-1)^3-3(-1)=2
\]
\[
f(1)=1^3-3(1)=-2
\]
So:
\[
\boxed{\text{local maximum at }(-1,2)}
\]
\[
\boxed{\text{local minimum at }(1,-2)}
\]
5. Second derivative check for the example
The second derivative is:
\[
f''(x)=6x
\]
At \(x=-1\):
\[
f''(-1)=-6<0
\]
So \(x=-1\) is a local maximum.
At \(x=1\):
\[
f''(1)=6>0
\]
So \(x=1\) is a local minimum.
6. When the second derivative test fails
Sometimes \(f''(c)=0\), so the second derivative test gives no conclusion.
For example:
\[
f(x)=x^3
\]
\[
f'(x)=3x^2
\]
\[
f'(0)=0
\]
\[
f''(x)=6x
\]
\[
f''(0)=0
\]
The second derivative test is inconclusive. The first derivative sign chart shows
that \(f'(x)\) stays positive on both sides of \(0\), so \(x=0\) is neither a local
maximum nor a local minimum.
7. Step-by-step method
- Find \(f'(x)\).
- Solve \(f'(x)=0\), and also check where \(f'(x)\) is undefined.
- Build a sign chart for \(f'(x)\).
- Use sign changes in \(f'(x)\) to classify each critical point.
- Find \(f''(x)\).
- Evaluate \(f''(c)\) at each critical point.
- Use the second derivative test as a confirmation or shortcut when possible.
8. Common mistakes
- Finding \(f'(x)\) but forgetting to solve \(f'(x)=0\).
- Thinking every critical point must be a maximum or minimum.
- Using \(f''(c)=0\) as if it proves “neither.” It only means the second derivative test is inconclusive.
- Forgetting to test signs on both sides of a critical point.
- Confusing local extrema with absolute extrema.
9. Main takeaway
\[
\boxed{f':+\to-\Rightarrow \text{local maximum}}
\]
\[
\boxed{f':-\to+\Rightarrow \text{local minimum}}
\]
\[
\boxed{f''(c)>0\Rightarrow \text{local minimum},\qquad f''(c)<0\Rightarrow \text{local maximum}}
\]
The first derivative test is usually the most reliable because it directly checks
whether the function changes from increasing to decreasing or from decreasing to increasing.
