Standard +0.3 This is a straightforward application of the quotient rule to find dy/dx, setting it equal to zero, and solving for x. The algebra involves exponentials but is routine for P3 level. Finding the y-coordinate requires simple substitution. Slightly above average difficulty due to the exponential manipulation required, but still a standard textbook exercise with no novel insight needed.
4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 + \mathrm { e } ^ { 3 x } }\) has one stationary point. Find the exact values of the coordinates of this point.
Equate derivative to zero and obtain a horizontal equation
M1
Carry out complete method for solving an equation of the form \(ae^{3x} = b\), or \(ae^{5x} = be^{2x}\)
M1
Obtain \(x = \ln 2\), or exact equivalent
A1
Obtain \(y = \frac{1}{3}\), or exact equivalent
A1
6
Use correct quotient or product rule | M1 |
Obtain correct derivative in any form | A1 |
Equate derivative to zero and obtain a horizontal equation | M1 |
Carry out complete method for solving an equation of the form $ae^{3x} = b$, or $ae^{5x} = be^{2x}$ | M1 |
Obtain $x = \ln 2$, or exact equivalent | A1 |
Obtain $y = \frac{1}{3}$, or exact equivalent | A1 | 6 |
4 The curve with equation $y = \frac { \mathrm { e } ^ { 2 x } } { 4 + \mathrm { e } ^ { 3 x } }$ has one stationary point. Find the exact values of the coordinates of this point.
\hfill \mbox{\textit{CAIE P3 2015 Q4 [6]}}