C3 January 2009 Q5
5. The functions f and g are defined by\[\mathrm{f} : x \mapsto 3x + \ln x, \quad x > 0, \quad x \in \mathbb{R}\]\[\mathrm{g} : x \mapsto \mathrm{e}^{x^2}, \quad x \in \mathbb{R}\]
(a) Write down the range of g. (1)
(b) Show that the composite function fg is defined by\[\mathrm{fg} : x \mapsto x^2 + 3\mathrm{e}^{x^2}, \quad x \in \mathbb{R}.\] (2)
(c) Write down the range of fg. (1)
(d) Solve the equation \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\mathrm{fg}(x)\right] = x(x\mathrm{e}^{x^2} + 2)\). (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{g}(x) \geqslant 1\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\mathrm{fg}(x) = \mathrm{f}\left(\mathrm{e}^{x^2}\right) = 3\mathrm{e}^{x^2} + \ln\mathrm{e}^{x^2}\) | M1 |
| \(= x^2 + 3\mathrm{e}^{x^2}\ \ \ast\) \(\left(\mathrm{fg} : x \mapsto x^2 + 3\mathrm{e}^{x^2}\right)\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathrm{fg}(x) \geqslant 3\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(x^2 + 3\mathrm{e}^{x^2}\right) = 2x + 6x\mathrm{e}^{x^2}\) | M1 A1 |
| \(2x + 6x\mathrm{e}^{x^2} = x^2\mathrm{e}^{x^2} + 2x\) \(\mathrm{e}^{x^2}(6x - x^2) = 0\) | M1 |
| \(\mathrm{e}^{x^2} \neq 0\), \(6x - x^2 = 0\) | A1 |
| \(x = 0, 6\) | A1 A1 |
| (6) | |
| (10 marks) |