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Question
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Options
True
False
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Solution
True.
Explanation:
f(x) = `"x"*"e"^("x" (1 - "x"))`
∴ f '(x) = `"e"^("x" (1 - "x")) + "x"*"e"^("x" (1 - "x")) [1 - 2"x"]`
`= "e"^("x" (1 - "x")) [1 + "x" - 2"x"^2]`
If f(x) is increasing, then f '(x) > 0.
Consider f '(x) > 0
∴ `"e"^("x" (1 - "x")) (1 + "x" - 2"x"^2)` > 0
∴ 2x2 - x - 1 < 0
∴ (2x + 1)(x - 1) < 0
ab < 0 ⇔ a > 0 and b < 0 or a < 0 or b > 0
∴ Either (2x + 1) > 0 and (x – 1) < 0 or
(2x + 1) < 0 and (x – 1) > 0
Case 1: (2x + 1) > 0 and (x – 1) < 0
∴ x > `-1/2` and x < 1
i.e., x ∈ `(-1/2, 1)`
Case 2: (2x + 1) < 0 and (x – 1) > 0
∴ x < `- 1/2` and x > 1
which is not possible.
∴ f(x) is increasing on `(-1/2, 1)`
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