Mathematics · Differentiation · 8 marks
A curve has equation y = x²e⁻ˣ. (a) Find dy/dx. [3] (b) Find the coordinates of the two stationary points, and determine the nature of each. [5]
Mark scheme
- (a) Product rule with u = x², v = e⁻ˣ
- (a) dy/dx = 2xe⁻ˣ + x²(−e⁻ˣ)
- (a) dy/dx = xe⁻ˣ(2 − x)
- (b) Setting dy/dx = 0: x = 0 or x = 2, since e⁻ˣ ≠ 0
- (b) Stationary points at (0, 0) and (2, 4e⁻²)
- (b) d²y/dx² = e⁻ˣ(x² − 4x + 2)
- (b) At x = 0: d²y/dx² = 2 > 0, a minimum
- (b) At x = 2: d²y/dx² = −2e⁻² < 0, a maximum
Factorise dy/dx before solving — it makes both roots obvious and stops you dividing by e⁻ˣ and losing one. Stating that e⁻ˣ is never zero is itself a mark in some schemes.