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Question:

Consider the following statements:

(i) When 0 < x < 1, then $\frac{1}{1 + x}$ < 1 - x + x²

(ii) When 0 < x < 1, then $\frac{1}{1 + x}$ > 1 - x + x²

(iii) When -1 < x < 0, then $\frac{1}{1 + x}$ < 1 - x + x²

(iv) When -1 < x < 0, then $\frac{1}{1 + x}$ > 1 - x + x²

Then the correct statements are

Explanation:

When 0 < x < 1, consider $\frac{1}{1 + x}$ and 1 - x + x²
Let us take x = 0.5
∴ $\frac{1}{1 + x}$ = 0.67 and 1 - x + x² = 1 - 0.5 + 0.25 = 0.75
⇒ $\frac{1}{1 + x}$ < 1 - x + x²
∴ (i) is correct.

When 0 < x < 1, consider $\frac{1}{1 + x}$ and 1 - x + x²
Let us take x = -0.5
∴ $\frac{1}{1 + x}$ = 2 and 1 - x + x² = 1 + 0.5 + 0.25 = 1.75
⇒ $\frac{1}{1 + x}$ > 1 - x + x²
∴ (iv) is correct.

Hence, option (d).

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