If \(a, b, c\) are positive real numbers, then the least value of \((a + b + c)(ab + bc + ca)\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(9abc\)
By the AM-GM inequality for positive numbers:
\(a + b + c \ge 3(abc)^{1/3}\)
\(ab + bc + ca \ge 3(ab \cdot bc \cdot ca)^{1/3} = 3(abc)^{2/3}\)
Both sides of each inequality are positive, so the inequalities can be multiplied:
\((a + b + c)(ab + bc + ca) \ge 9(abc)^{1/3 + 2/3}\)
\(= 9abc\)
Equality holds when a = b = c (for example, a = b = c = 1 gives \(3 \times 3 = 9 = 9abc\)), so the least value is 9abc.