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

Answer 1

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)`