A decreasing exponential curve with a stationary point

A decreasing exponential curve with a stationary point

Secondary 4
8 marks
TGM Original Questions

A curve has equation \(y=(x^2+2x+2)e^{-x}\), where \(x\) is real.

  1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=-x^2e^{-x}\).[3]
  2. Find the coordinates of its stationary point and explain why it is neither a maximum nor a minimum.[2]
  3. Use the second derivative near the stationary point to show that it is a stationary point of inflexion.[3]

Solution:

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Answer:(b) \((0,2)\); derivative negative on both sides. (c) \(y''=x(x-2)e^{-x}\), changing from positive to negative at \(x=0\).

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