Explanation

(b)
$f(x) = 1 + 2x^{2} + 2^{2}x^{4} + ... + 2^{10}x^{20}$
$f^{'}(x) = 4x + {4.2}^{2}x^{3} + ... + {20.2}^{10}x^{19}$
$= x(4 + {4.2}^{2}x^{2} + ... + {20.2}^{10}x^{18})$
For a maximum or minimum, put $f^{'}(x) = 0$
$\Rightarrow \ \ \ \ \ x = 0$
But $4 + {12.2}^{2} + x^{2} + \ldots + {20.19.2}^{10}x^{18} > 0$
For $x < 0\ \ \ \Rightarrow \ \ \ \ f^{'}(x) < 0$ and $x > 0\ \ \Rightarrow \ \ \ \ \ f^{'}(x) > 0$
$\therefore$ Exactly one minimum.