Theory of Intelligence
A-LEVEL•Mathematics•Quadratics•Question 5

Solution

🟡 Exam-style · [5 marks] The function $f$ is defined by $f(x) = -x^2 + 6x - 4$ for $x \in \mathbb{R}$.

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Solution

(a)

−x2+6x−4=−[x2−6x]−4=−[(x−3)2−9]−4=5−(x−3)2\begin{aligned} -x^2 + 6x - 4 &= -\left[x^2 - 6x\right] - 4 \\ &= -\left[(x-3)^2 - 9\right] - 4 \\ &= 5 - (x-3)^2 \end{aligned}

(b) The greatest value is 55, when x=3x = 3.

(c) The vertex x=3x = 3 lies inside 0≤x≤40 \le x \le 4, so the greatest value is still 55. Check the endpoints:

  • f(0)=5−9=−4f(0) = 5 - 9 = -4
  • f(4)=5−1=4f(4) = 5 - 1 = 4

The smallest value is −4-4, so the range is −4≤f(x)≤5-4 \le f(x) \le 5.

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