# State whether the following statement is True or False: The function f(x) = x⋅ex(1-x) is increasing on (-12,1). - Mathematics and Statistics

MCQ
True or False

State whether the following statement is True or False:

The function f(x) = "x"*"e"^("x" (1 - "x")) is increasing on ((-1)/2, 1).

• True

• False

#### 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)

Concept: Increasing and Decreasing Functions
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 4 Applications of Derivatives
Miscellaneous Exercise 4 | Q 3.4 | Page 114