Turning points and inflexions of an exponential product
Turning points and inflexions of an exponential product
IP 4
10 marks
A curve has equation \(y=x^2e^{-x}\).
- Show that \(\dfrac{\mathrm dy}{\mathrm dx}=x(2-x)e^{-x}\). Hence find the coordinates of the stationary points and determine their nature.[4]
- Show that \(\dfrac{\mathrm d^2y}{\mathrm dx^2}=\left(x^2-4x+2\right)e^{-x}\), and find the exact \(x\)-coordinates of the points of inflexion.[3]
- Explain why \(y\ge0\) for all \(x\), and state the number of real solutions of \(x^2e^{-x}=k\) when \(0<k<4e^{-2}\), justifying your answer.[3]
Solution:
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Answer:(a) minimum \((0,0)\), maximum \(\left(2,4e^{-2}\right)\) (b) \(x=2\pm\sqrt2\) (c) \(y\ge0\) since \(x^2\ge0\) and \(e^{-x}>0\); \(3\) solutions
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