A-LevelMathematicsDifferentiation9709/23 · May/June 2023 · Q59 Marks
The diagram shows the curve with equation y = e⁻½ˣ(x² – 5x + 4). The curve crosses the x-axis at the points A and B, and has a maximum at the point C.
- (a)Find the exact gradient of the curve at B.[5]
- (b)Find the exact coordinates of C.[4]
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Mark scheme
Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.
- [*M1] Attempt use of product rule to find first derivative
- [A1] Obtain -½e⁻½ˣ(x²-5x+4)+e⁻½ˣ(2x-5)
- [B1] Obtain x = 4 for point B
- [DM1] Substitute x=4 to find the value of the derivative
- [A1] Obtain 3e⁻²
- [*M1] Equate their first derivative to zero and simplify as far as quadratic equation
- [A1] Obtain at least x²-9x+14 = 0
- [DM1] Solve to find relevant x value and substitute to find the value of y
- [A1] Obtain x = 7 and y=18e⁻³·⁵
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