- Difficulty
- Capstone
- Suggested time
- 9 min
- Marks
- 5
- Theory
- CLP-1 §2.13Key results
The Mean Value Theorem practice question 10
For which values of the constant does have exactly one real root? Enter the smallest positive integer with this property.
How many times a cubic crosses the axis depends on where its turning points sit relative to it. Where are the turning points of , and how does changing move the graph?
vanishes at , so by Rolle’s theorem has at most three roots, and its shape is: rise to a local maximum at , fall to a local minimum at , rise again.
crosses the axis exactly once precisely when the local maximum and the local minimum have the same sign. Compute and in terms of .
A whole number.
Theory CLP-1 §2.13Key results
Worked solution
Strategy
The two critical values control everything. A cubic with a local max and a local min has three roots when they straddle the axis, two when one sits on it, and one when both are on the same side. Express the condition in k.
Working
Critical points at −1 and 1; f increases, decreases, then increases.
The local maximum value and the local minimum value.
By the IVT and monotonicity on each piece: three roots if k − 2 < 0 < k + 2, i.e. −2 < k < 2; a repeated root if k = ±2; exactly one root if both critical values have the same sign, i.e. k > 2 or k < −2.
k = 3 gives f(−1) = 5 and f(1) = 1, both positive, so the graph crosses the axis only once, to the left of −1.
Answer
The answer is exactly one root when the absolute value of k exceeds 2; the smallest positive integer is 3.
Check
x³ − 3x + 3 has its only real root near x = −2.10, while x³ − 3x + 2 = (x − 1)²(x + 2) has two.
Where this goes wrong
Answering k = 2. At k = 2 the local minimum sits exactly on the axis, giving a double root at x = 1 and a second root at x = −2: two distinct roots, not one.
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