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Question
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
Solution
f(x) = xex `\implies` f'(x) = ex (x + 1)
When x ∈ [–1, ∞), (x + 1) ≥ 0 and ex > 0
`\implies` f'(x) ≥ 0
∴ f(x) increases in this interval.
or, we can write f(x) = xex `\implies` f'(x) = ex (x + 1)
For f(x) to be increasing, we have f'(x) = ex (x + 1) ≥ 0 `\implies` x ≥ –1 as ex > 0, ∀ x ∈ R
Hence, the required interval where f(x) increases is [–1, ∞).
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