- Topic
- Curve sketching
- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
Curve sketching practice question 22
A function , defined for all real , has derivative . You are given , not . At which value of does have a local maximum?
You never need itself. Find where is zero and make a sign chart for .
is zero at and , and , so its sign comes from . A local maximum is where that sign goes from positive to negative.
A whole number.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
The first-derivative test needs only the sign of f′. The exponential factor is always positive, so the sign is decided by the two linear factors.
Working
Critical points; e−x never vanishes.
Sign chart of (3 − x), (x + 1), e−x.
f′ goes − to + at −1 and + to − at 3.
Answer
The answer is 3.
Check
f″(3) = −4e−3 is negative, so the second derivative test also gives a local maximum at x = 3.
Where this goes wrong
Differentiating f′ again and setting f″ = 0, as if looking for an inflection point, or trying to integrate to recover f. The question only needs the sign of the derivative provided.
Try a similar question
Find the -coordinate of the local maximum of .Saved in this browser only.