Question

Which of the following inequalities is/are TRUE?

Answer 1

Answer:

A , B , D

Step-by-step explanation:

Solution is refer to the attachment

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Answer 2

which of the following inequalities is/are TRUE ?

solution : expansion of cosx ≈ 1 - x²/2 + x⁴/4 - x⁶/6 + ....

so, definitely , cosx ≥ 1 - x²/2

now ∫x cosx dx ≥ ∫x (1 - x²/2) dx = ∫(x - x³/2)dx

= [x²/2 - x⁴/8]

= [1/2 - 1/8] = 3/8

so, ∫₀¹x cosx dx ≥ 3/8 is correct option.

expansion of sinx = x - x³/3! + x⁵/5! - .... ∞

so definitely sinx ≥ x - x³/3! = x - x³/6

now ∫x sinx dx ≥ ∫x(x - x³/6) dx

= [x³/3 - x⁵/30 ]

= [1/3 - 1/30] = 3/10 > 2/9

so, ∫₀¹x sinx dx ≥ 2/9 is also correct option

now x²cosx ≥ x²[1 - x²/2] = x²- x⁴/2

∫₀¹ x² cosx dx ≥ ∫(x² - x⁴/2) dx = [x³/3 - x⁵/10]

= 1/3 - 1/10] = 7/30 < 1/2

so, option (c) is wrong.

x² sinx ≥ x² [x - x³/6 ] = [x³ - x⁵/6]

∫₀¹ x² sinx dx ≥ ∫₀¹ (x³ - x⁵/6) dx = [1/4 - 1/36] = 2/9

so option (D) is correct

Therefore options (A), (B) and (D) are correct choices.