- Topic
- Curve sketching
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Curve sketching practice question 11
Find the local maximum value of .
Give the value of the function, not the x-coordinate.
Find the critical points by solving , then decide which one is the local maximum.
. The derivative changes from positive to negative at a local maximum.
A whole number.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Locate the critical points, classify them with the first or second derivative test, then evaluate the original function at the one that is a maximum.
Working
Differentiate and factor. Critical points at x = −1 and x = 3.
The second derivative test is quicker than a sign chart here.
Negative second derivative means concave down, so this critical point is a peak.
Evaluate the original function, not the derivative, to get the value.
Answer
The answer is 10.
Check
f(−1.1) = 9.939 and f(−0.9) = 9.941 both fall just short of f(−1) = 10.
Where this goes wrong
Answering −1, which is where the maximum occurs rather than what the maximum value is. Read carefully whether the question wants the location or the value.
Leans on
Differentiating a polynomial, from Differentiation rules
Try a similar question
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