- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 3
- Theory
- CLP-1 §2.13Key results
The Mean Value Theorem practice question 9
How many real roots does the function below have?
Rolle’s theorem puts a root of between any two roots of . So how does the number of roots of cap the number of roots of ?
has a single real root. How many roots can have at most?
Now find out how many it actually has: check the signs of , and , and use the IVT on each side of the critical point.
A whole number.
Theory CLP-1 §2.13Key results
Worked solution
Strategy
Rolle’s theorem caps the number of roots at one more than the number of critical points. The IVT then shows the cap is reached.
Working
Exactly one critical point.
If p had three roots, Rolle’s theorem would give two distinct zeros of p′ between them. It has only one, so p has at most two roots.
Sign changes on (0, 1) and (1, 2): the IVT gives a root in each.
At most two and at least two.
Answer
The answer is exactly two.
Check
The roots are near 0.251 and 1.493, and there are no negative ones, since x⁴ − 4x + 1 ≥ 1 whenever x ≤ 0.
Where this goes wrong
Answering 4 because the degree is 4. The degree bounds the count from above; the actual number of real roots here is decided by the single turning point at x = 1 and the sign of p there.
Leans on
The Intermediate Value Theorem, from Continuity
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