TS EAMCET · Maths · Inverse Trigonometric Functions
The range of the real valued function \(f(x)=\operatorname{Sin}^{-1}\left(\sqrt{x^2+x+1}\right)\) is
- A \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
- B \(\left[0, \frac{\pi}{2}\right]\)
- C \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\)
- D \(\left[\frac{\pi}{3}, \frac{\pi}{2}\right]\)
Answer & Solution
Correct Answer
(D) \(\left[\frac{\pi}{3}, \frac{\pi}{2}\right]\)
Step-by-step Solution
Detailed explanation
Let \( g(x) = x^2+x+1 \). Minimum value of \( g(x) \) occurs at \( x = -1/(2 \cdot 1) = -1/2 \). \( g(-1/2) = (-1/2)^2 + (-1/2) + 1 = 1/4 - 1/2 + 1 = 3/4 \) So, \( x^2+x+1 \ge 3/4 \). For \( \operatorname{Sin}^{-1}(y) \) to be defined, \( -1 \le y \le 1 \). Thus,…
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