Which of the following inequalities are NOT correct?
(A) If $a >1,b> 1$, then $\log_ba+ \log_a b ≤2$
(B) For any real number $x, (9^x +9^{1-x}) ≥9$
(C) If a, b, c are non-zero real numbers of the same sign, then $\frac{a}{b}+\frac{b}{c} +\frac{c}{a}≤3$
(D) If a, b, c are three distinct real numbers, then $(a + b)(b+c)(c+a) > 8abc$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
• (A) is false in general because $\frac{\ln a}{\ln b}+\frac{\ln b}{\ln a}\ge 2$, not $\le 2$.
• (B) is false because the minimum of $9^{x}+9^{1-x}$ is $6$, which is not $\ge 9$.
• (C) is false because $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge 3$, not $\le 3$.
• (D) is true for three distinct real numbers (with positive values): $(a+b)(b+c)(c+a)>8abc$.
Thus, the NOT correct statements are (A), (B), and (C).