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Question
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
Solution
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)]`
∵ ex 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, ∞)
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