Problem 29 · 2021 Math Kangaroo
Stretch
Algebra & Patterns
substitutioncasework
Let \(M(k)\) be the maximum value of \(\left|\,4x^{2} - 4x + k\,\right|\) for x in the interval \([-1,1]\), where k can be any real number. What is the minimum possible value of \(M(k)\)?
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Answer: B — \(\tfrac{9}{2}\)
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Hint 1 of 2
Find the range of g(x) = 4x² − 4x on [−1,1]; then 4x² − 4x + k just shifts that range by k.
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Hint 2 of 2
The maximum of the absolute value is the larger of the two endpoint distances; choose k to balance them.
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Approach: find the range, then minimise the larger absolute endpoint
- On [−1,1], g(x)=4x²−4x ranges from −1 (at x=½) to 8 (at x=−1), so 4x²−4x+k lies in [k−1, k+8].
- Thus M(k) = max(|k−1|, |k+8|); this is smallest when k−1 = −(k+8), i.e. k = −7/2.
- Then both equal 9/2, the minimum value of M(k).
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