Date | May 2008 | Marks available | 7 | Reference code | 08M.2.hl.TZ2.6 |
Level | HL only | Paper | 2 | Time zone | TZ2 |
Command term | Find and Justify | Question number | 6 | Adapted from | N/A |
Question
Consider the curve with equation f(x)=e−2x2 for x<0 .
Find the coordinates of the point of inflexion and justify that it is a point of inflexion.
Markscheme
METHOD 1
EITHER
Using the graph of y=f′(x) (M1)
A1
The maximum of f′(x) occurs at x = −0.5 . A1
OR
Using the graph of y=f″. (M1)
A1
The zero of f''(x) occurs at x = −0.5 . A1
THEN
Note: Do not award this A1 for stating x = ±0.5 as the final answer for x .
f( - 0.5) = 0.607( = {{\text{e}}^{ - 0.5}}) A2
Note: Do not award this A1 for also stating (0.5, 0.607) as a coordinate.
EITHER
Correctly labelled graph of f'(x) for x < 0 denoting the maximum f'(x) R1
(e.g. f'( - 0.6) = 1.17 and f'( - 0.4) = 1.16 stated) A1 N2
OR
Correctly labelled graph of f''(x) for x < 0 denoting the maximum f'(x) R1
(e.g. f''( - 0.6) = 0.857 and f''( - 0.4) = - 1.05 stated) A1 N2
OR
f'(0.5) \approx 1.21. f'(x) < 1.21 just to the left of x = - \frac{1}{2}
and f'(x) < 1.21 just to the right of x = - \frac{1}{2} R1
(e.g. f'( - 0.6) = 1.17 and f'( - 0.4) = 1.16 stated) A1 N2
OR
f''(x) > 0 just to the left of x = - \frac{1}{2} and f''(x) < 0 just to the right of x = - \frac{1}{2} R1
(e.g. f''( - 0.6) = 0.857 and f''( - 0.4) = - 1.05 stated) A1 N2
[7 marks]
METHOD 2
f'(x) = - 4x{{\text{e}}^{ - 2{x^2}}} A1
f''(x) = - 4{{\text{e}}^{ - 2{x^2}}} + 16{x^2}{{\text{e}}^{ - 2{x^2}}}\,\,\,\,\,\left( { = (16{x^2} - 4){{\text{e}}^{ - 2{x^2}}}} \right) A1
Attempting to solve f''(x) = 0 (M1)
x = - \frac{1}{2} A1
Note: Do not award this A1 for stating x = \pm \frac{1}{2} as the final answer for x .
f\left( { - \frac{1}{2}} \right) = \frac{1}{{\sqrt {\text{e}} }}{\text{ }}( = 0.607) A1
Note: Do not award this A1 for also stating \left( {\frac{1}{2},\frac{1}{{\sqrt {\text{e}} }}} \right) as a coordinate.
EITHER
Correctly labelled graph of f'(x) for x < 0 denoting the maximum f'(x) R1
(e.g. f'( - 0.6) = 1.17 and f'( - 0.4) = 1.16 stated) A1 N2
OR
Correctly labelled graph of f''(x) for x < 0 denoting the maximum f'(x) R1
(e.g. f''( - 0.6) = 0.857 and f''( - 0.4) = - 1.05 stated) A1 N2
OR
f'(0.5) \approx 1.21. f'(x) < 1.21 just to the left of x = - \frac{1}{2}
and f'(x) < 1.21 just to the right of x = - \frac{1}{2} R1
(e.g. f'( - 0.6) = 1.17 and f'( - 0.4) = 1.16 stated) A1 N2
OR
f''(x) > 0 just to the left of x = - \frac{1}{2} and f''(x) < 0 just to the right of x = - \frac{1}{2} R1
(e.g. f''( - 0.6) = 0.857 and f''( - 0.4) = - 1.05 stated) A1 N2
[7 marks]
Examiners report
Most candidates adopted an algebraic approach rather than a graphical approach. Most candidates found f'(x) correctly, however when attempting to find f''(x), a surprisingly large number either made algebraic errors using the product rule or seemingly used an incorrect form of the product rule. A large number ignored the domain restriction and either expressed x = \pm \frac{1}{2} as the x-coordinates of the point of inflection or identified x = \frac{1}{2} rather than x = - \frac{1}{2}. Most candidates were unsuccessful in their attempts to justify the existence of the point of inflection.