Explanation

(a)
Given that, $a^{2} + b^{2} + c^{2} = 1\ \ \ \ \ \$…(i)
Now, $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca) \geq 0$
$\Rightarrow 2(ab + bc + ca) \geq - 1\ \ \lbrack$from Eq.(i)]
$\Rightarrow ab + bc + ca \geq - \frac{1}{2}$ …(iii)
Also, $a^{2} + b^{2} + c^{2} - ab - bc - ca$
$= \frac{1}{2}\{(a - b)^{2} + (b - c)^{2} + (c - a)^{2}\} \geq 0$
$\Rightarrow ab + bc + ca \leq a^{2} + b^{2} + c^{2}$
$\Rightarrow ab + bc + ca \leq 1\ \ \ \$[from Eq.(i)] …(iii)
From relation (ii) and (iii), we get
$- \frac{1}{2} \leq ab + bc + ca \leq 1$