Question

Find the interval in which the function f(x) = x²e^–x is strictly increasing or decreasing.

Answer 1

f(x) = x²e^–x

`dy/dx = -x^2e^-x + e^-x * 2x`

= `1/e^x(2x - x^2)`

= `1/e^x(2 - x)x`

= `-e^-x[x(x - 2)]`

∵ e^x is always positive

∴ Setting x(x – 2) = 0

`\implies` x = 0 or x = 2

For strictly decreasing function

`dy/dx < 0`

`\implies` x < 0 or x > 2

`\implies` x ∈ (– ∞, 0) ∪ (2, ∞)

For strictly increasing function

`dy/dx > 0`

`\implies` 0 < x < 2

`\implies` x ∈ (0, 2)

∴ f(x) is increasing in (0, 2) and decreasing in (– ∞, 0) ∪ (2, ∞)