The largest interval lying in (-(π/2), (π/2)) for which the function f (x)=4-x2+cos-1((x/2)-1)+log(cos x) is defined, is

Question

The largest interval lying in (-(π/2), (π/2)) for which the function f (x)=4-x2+cos-1((x/2)-1)+log(cos x) is defined, is (A) [0, π] (B) (-(π/2), (π/2)) (C) [-(π/4), (π/2)) (D) [0, (π/2))

Tardigrade Admin · Accepted Answer

f (x) is defined if -1 le (x/2)-1 le1 and cos x > 0 or 0 le x le 4 and -(π/2) < x < (π/2) ∴ x ∈0, [(π/2))

Q. The largest interval lying in $(- \frac{π}{2}, \frac{π}{2})$ for which the function $f (x) = 4^{- x^{2}} + co s^{- 1} (\frac{x}{2} - 1) + l o g (cos x)$ is defined, is

$[0, π]$

$(- \frac{π}{2}, \frac{π}{2})$

$[- \frac{π}{4}, \frac{π}{2})$

$[0, \frac{π}{2})$

$f (x)$ is defined if $- 1 \leq \frac{x}{2} - 1 \leq 1$ and $cos x > 0$
or $0 \leq x \leq 4$ and $- \frac{π}{2} < x < \frac{π}{2}$
$∴ x \in 0, [\frac{π}{2})$

Q. The largest interval lying in (−2π​,2π​) for which the function f(x)=4−x2+cos−1(2x​−1)+log(cosx) is defined, is

[0,π]

(−2π​,2π​)

[−4π​,2π​)

[0,2π​)