- Topic
- Optimization
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Optimization practice question 2
Find the coordinates of the local minimum of .
Each choice is a point. Test it twice: is its a critical point where switches from falling to rising, and is its second coordinate the value of there?
Critical points are where . Then decide which one is the minimum, using the sign of on either side or the second derivative.
Theory UBC Math 100 §8CLP-1 §3.5Key results
Worked solution
Strategy
Find the critical points, classify them with the second derivative, and evaluate h at the minimum to get both coordinates.
Working
Two critical points.
Second derivative test.
The local minimum is at (3, −49); the local maximum is at (−3, 59).
Answer
The answer is the point 3, negative 49.
Check
h(2) = −41 and h(4) = −39 both lie above h(3) = −49, so the curve dips to a low point at (3, −49).
Where this goes wrong
Reporting (−3, 59), the local maximum, or (3, 5) by forgetting to evaluate h. A local minimum is a point on the curve, with the function value as its second coordinate.
Leans on
The second derivative test, from Curve sketching
Try a similar question
Find the local maximum value of .Saved in this browser only.