November 2021 Paper 2 Q18
18
\[y = 12x + \frac{3}{x}\]Show that \(y\) has a minimum value when \(\quad x = 0.5\) [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| 12 or \(-3x^{-2}\) | M1 | oe eg \(12x^0\) or \(3 \times -1x^{-1-1}\) or \(-\dfrac{3}{x^2}\) |
| 12 and \(-3x^{-2}\) | M1dep | oe eg \(12 - \dfrac{3}{x^2}\) or \(12x^0\) and \(3 \times -1x^{-1-1}\) |
| \(12 - 3x^{-2} = 0\) and \(x = 0.5\) or \(12 - 3 \times 0.5^{-2} = 0\) | M1dep | oe \(= 0\) must be seen condone inclusion of \(x = -0.5\) |
| \(6x^{-3}\) | M1 | oe eg \(-2 \times -3x^{-2-1}\) ft differentiation of their first derivative if it involves a negative power of \(x\) |
| M4 and \(6 \times 0.5^{-3}\ (= 48)\) which is positive (so minimum) | A1 | oe do not allow if \(\dfrac{6}{0.5^3}\) is evaluated incorrectly |
| Alternative method 2 | ||
| 12 or \(-3x^{-2}\) | M1 | oe eg \(12x^0\) or \(3 \times -1x^{-1-1}\) |
| 12 and \(-3x^{-2}\) | M1dep | oe eg \(12 - \dfrac{3}{x^2}\) or \(12x^0\) and \(3 \times -1x^{-1-1}\) |
| \(12 - 3x^{-2} = 0\) and \(x = 0.5\) or \(12 - 3 \times 0.5^{-2} = 0\) | M1dep | oe \(= 0\) must be seen condone inclusion of \(x = -0.5\) |
| Substitutes one \(x\) value in range (0, 0.5) into \(12 - 3x^{-2}\) and substitutes one \(x\) value \(\gt 0.5\) into \(12 - 3x^{-2}\) | M1 | eg \(12 - 3 \times 0.25^{-2}\) and \(12 - 3 \times 1^{-2}\) ft substitution into their first derivative if it involves a negative power of \(x\) |
| M4 and two correct evaluations (so minimum) or M4 and two correct signs shown with no incorrect evaluations (so minimum) | A1 | eg M4 and \(12 - 3 \times 0.25^{-2} = -36\) and \(12 - 3 \times 1^{-2} = 9\) (so minimum) or M4 and \(12 - 3 \times 0.25^{-2}\) is negative and \(12 - 3 \times 1^{-2}\) is positive (so minimum) |
Additional guidance
| Alt 1 \(12 + 3x^{-2} = 0\) \(-6x^{-3}\) | M1M0M0 M1A0 |
| Alt 2 \(12 - 3x^{-2}\) \(6x^{-3}\) \(12 - 3 \times 0.25^{-2} = -36 \qquad 12 - 3 \times 1^{-2} = 9\) so minimum (A1 only possible after awarding M4) | M1M1M0 M1 A0 |
| Ignore any testing of the stationary point at \(x = -0.5\) |