## Question 3:

### Part (a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 2\sqrt{x} + \frac{18}{\sqrt{x}} - 1 = 2x^{\frac{1}{2}} + 18x^{-\frac{1}{2}} - 1$ | — | — |
| $\frac{dy}{dx} = x^{-\frac{1}{2}} - 9x^{-\frac{3}{2}}$ | M1A1A1 | M1: $x^n \rightarrow x^{n-1}$. A1: $x^{-\frac{1}{2}}$. A1: $-9x^{-\frac{3}{2}}$ and $-1 \rightarrow 0$ |

### Part (a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{d^2y}{dx^2} = -\frac{1}{2}x^{-\frac{3}{2}} + \frac{27}{2}x^{-\frac{5}{2}}$ | M1A1ft | M1: $x^n \rightarrow x^{n-1}$. A1: $-\frac{1}{2}x^{-\frac{3}{2}} + \frac{27}{2}x^{-\frac{5}{2}}$ |

### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x^{-\frac{1}{2}} - 9x^{-\frac{3}{2}} = 0$ | M1 | Set $\frac{dy}{dx} = 0$ and proceed to $x=$ |
| $x - 9 = 0 \Rightarrow x = 9$ **only** | A1 | Cso |
| $y = 2\sqrt{"9"} + \frac{18}{\sqrt{"9"}} - 1$ | M1 | Substitutes their $x$ value(s) into the given equation |
| $y = 11$ | A1 | Cao. There must be no other turning points for this mark but allow recovery if $x=9$ is obtained by the invalid method shown below |

### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f''(9) = -\frac{1}{2}("9")^{-\frac{3}{2}} + \frac{27}{2}("9")^{-\frac{5}{2}}$ | M1 | Substitutes their $x$ value(s) into their second derivative |
| $f''(9) = \frac{1}{27} > 0 \therefore$ Minimum | A1 | Fully correct solution including correct numerical second derivative (awrt 0.04) and reference to positive or $> 0$. There must be no other turning points; $x=9$ only used but allow recovery as above |

---