Question

On which of the following intervals is the function x¹⁰⁰ + sin x – 1 decreasing ?

• A. (0, p/2)
• B. (0, 1)
• C. (p/2, p)
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

ƒ(x) = x¹⁰⁰ + sin x – 1 ̃ ƒ¢(x) = 100 x⁹⁹ + cos x.

If 0 < x < p/2, then ƒ¢(x) > 0, therefore ƒ(x) is increasing on (0, p/2).

If 0 < x < 1, then 100 x⁹⁹ + cos x > 0

[Q x lies between 0 and 1 radian]

̃ ƒ¢(x) = 100 x⁹⁹ + cos x > 0

̃ ƒ (x) is increasing on (0, 1).

If p/2 < x < p, then 100 x⁹⁹ > 100

[Q x > 1, \ x⁹⁹ > 1]

̃ 100 x⁹⁹ + cos x > 0

[Q cos x ³ –1, \ 100 x⁹⁹ + cos x > 99]

̃ ƒ¢(x) > 0 ̃ f(x) is increasing on (p/2, p).