Question

On which of the following intervals is the function x¹⁰⁰ + sin x – 1 strictly increasing?
(i) (1, 1) (ii) (0, 1) (iii) $open parentheses straight pi over 2 comma space straight pi close parentheses$ (iv) $open parentheses 0 comma space space straight pi over 2 close parentheses$

Solution

Let f (x) = x¹⁰⁰ + sin x – 1 ∴ f ' (x) = 100 x⁹⁹ + cos x
(i) For – 1 < x < 1, f (x) > 0 is not necessarily true
∴ f (x) is not strictly increasing on (– 1, 1)
(ii) For 0 < x < 1, f ' (x) > 0 ∴ f (x) is strictly increasing on (0, 1).
(iii) $1 space or space space straight pi over 2 space less than space straight x thin space less than space straight pi comma space space space straight f apostrophe left parenthesis straight x right parenthesis space greater than space 0 space space space space space space space therefore space space space space straight f left parenthesis straight x right parenthesis space is space strictly space increasing space on space open parentheses straight pi over 2 comma space straight pi close parentheses$
(iv) $For space 0 space less than space straight x less than space straight pi over 2 comma space space space straight f apostrophe left parenthesis straight x right parenthesis thin space greater than 0 space space space space space space space therefore space space space space straight f left parenthesis straight x right parenthesis space is space strictly space increasing space on space open parentheses 0 comma space space straight pi over 2 close parentheses.$

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Application Of Derivatives

On which of the following intervals is the function x¹⁰⁰ + sin x – 1 strictly increasing?
(i) (1, 1) (ii) (0, 1) (iii) $open parentheses straight pi over 2 comma space straight pi close parentheses$ (iv) $open parentheses 0 comma space space straight pi over 2 close parentheses$

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For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

On which of the following intervals is the function x100 + sin x – 1 strictly increasing?(i) (1, 1) (ii) (0, 1) (iii) (iv)

Some More Questions From Application of Derivatives Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

On which of the following intervals is the function x¹⁰⁰ + sin x – 1 strictly increasing?
(i) (1, 1) (ii) (0, 1) (iii) $open parentheses straight pi over 2 comma space straight pi close parentheses$ (iv) $open parentheses 0 comma space space straight pi over 2 close parentheses$