How to Find Maxima and Minima
To find the local maxima and minima of a function 
, follow these steps:
Step 1 — Find the first derivative 
.
Step 2 — Set 
and solve for 
. These are the stationary points (also called critical points).
Step 3 — Find the second derivative 
.
Step 4 — Classify each stationary point using the second derivative test:
- If
, the point is a local minimum. - If
, the point is a local maximum. - If
, the test is inconclusive — use the first derivative (sign change) test instead.
Step 5 — Find the y-coordinates by substituting each -value back into the original function .
Worked Problems and Solutions
Exercise 1: 
Solution of Exercise 1
First derivative:
Set :
Second derivative:
Classify:
At 
: 
> 0, so this is a local minimum.
At 
: 
< 0, so this is a local maximum.
Y-coordinates:
Conclusion: Local maximum at 
. Local minimum at 
.
Exercise 2: 
Solution of Exercise 2
First derivative:
Set :
Second derivative:
Classify:
At 
: 
< 0, so this is a local maximum.
At 
: 
> 0, so this is a local minimum.
Y-coordinates:
Conclusion: Local maximum at 
. Local minimum at 
.
Exercise 3: 
Solution of Exercise 3
First derivative:
Set :
Second derivative:
Classify:
At : 
< 0, so this is a local maximum.
At : 
> 0, so this is a local minimum.
At : 
> 0, so this is a local minimum.
Y-coordinates:
Conclusion: Local maximum at 
. Local minima at 
and 
.
Exercise 4: 
Solution of Exercise 4
First derivative (using the quotient rule):
Set :
The denominator is always positive, so set the numerator equal to zero:
Classify using the first derivative sign test:
For : we have 
> 0 and 
< 0. The derivative changes from positive to negative, so this is a local maximum.
For : we have 
< 0 and [latex]f'(0) = 1[/latex] > 0. The derivative changes from negative to positive, so this is a local minimum.
Y-coordinates:
Conclusion: Local maximum at 
. Local minimum at 
.
Exercise 5: 
Solution of Exercise 5
First derivative:
Set :
Second derivative:
Classify:
At : 
< 0, so this is a local maximum.
At 
: 
> 0, so this is a local minimum.
Y-coordinates:
Conclusion: Local maximum at 
. Local minimum at 
.
Exercise 6: 
Solution of Exercise 6
First derivative (using the product rule):
Set :
Since 
> 0 for all , we need:
Second derivative:
Classify:
At : 
< 0, so this is a local maximum.
Y-coordinate:
Conclusion: Local maximum at 
. There are no local minima.
Exercise 7: 
Solution of Exercise 7
Note: The domain is > 0.
First derivative:
Set :
Second derivative:
Classify:
At : 
< 0, so this is a local maximum.
Y-coordinate:
Conclusion: Local maximum at 
. There are no local minima on the domain > 0.
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