| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 11 |
| Topic | Differentiating Transcendental Functions |
| Type | Classify stationary points |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on exponential differentiation and stationary points. Part (i) requires differentiating e^(3x) - 5e^(2x) + 8e^x and solving the resulting equation (which factors nicely), part (ii) requires finding the second derivative and solving an inequality, and part (iii) applies the second derivative test. All steps are standard A-level techniques with no novel insight required, though it's slightly above average difficulty due to the algebraic manipulation needed when factoring expressions involving exponentials. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative |
**(i)** $\frac{dy}{dx} = 3e^{3x} - 10e^{2x} + 8e^x$ B1
Equating to zero: M1
Consider the solutions $r$ ($x = \ln r$) of $3r^2 - 10r + 8 = 0$ A1
Attempting to solve: M1
Obtain $r = 2$ or $4/3$
Stationary points $(\ln 2,\ 4)$ and $(\ln(4/3),\ 112/27)$ A1 **[5]**
**(ii)** $\frac{d^2y}{dx^2} = 9e^{3x} - 20e^{2x} + 8e^x$ B1
Consider the solutions of $9r^2 - 20r + 8 > 0$: M1
Obtain $r > 1.70$ OR $r < 0.523$ A1
Obtain $x > 0.530$ OR $x < -0.648$ A1 **[4]**
**(iii)** Correct use of the sign of the second derivative: M1
$2 > 1.70 \Rightarrow (\ln 2,\ 4)$ is a minimum
$0.523 < 4/3 < 1.70 \Rightarrow$ other point is a maximum A1 **[2]**9 A curve has equation
$$y = \mathrm { e } ^ { 3 x } - 5 \mathrm { e } ^ { 2 x } + 8 \mathrm { e } ^ { x }$$
(i) Find the exact coordinates of the stationary points of $y$.\\
(ii) Determine the range of values of $x$ for which
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } > 0$$
(iii) Determine the nature of the stationary points on the curve.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q9 [11]}}