Let the numbers \( 2, b, c \) be in an \( \mathrm{AP} \) and \( A=\...

Let the numbers \( 2, b, c \) be in an \( \mathrm{AP} \) and \( A=\left[\begin{array}{ccc}1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2}\end{array}\right] \).
If \( \operatorname{det}(A) \in[2,16] \), then \( c \) lies in the interval
(2019 Main, 8 April II)
(a) \( \left[3,2+2^{3 / 4}\right] \) (b) \( \left(2+2^{3 / 4}, 4\right) \) (c) \( [4,6] \)
(d) \( [2,3) \)