If a, b, c0, the minimum value of (a/b +c)+(b/c +a)+(c/a +b) is

Question

If a, b, c>0, the minimum value of (a/b +c)+(b/c +a)+(c/a +b) is (A) 1 (B)  (3/2)  (C) 2 (D)  (5/2)

Tardigrade Admin · Accepted Answer

Using AM ≥ GM ((a/b +c)+(b/c +a)+(c/a +b)/3) ≥ √[3](a b c/(a +b)(b +c)(c+ a)) ...(i) Again using AM ≥ GM (a +b/2) ≥ √a b, (b+c/2) ≥ √b c, (c +a/2) ≥ √c a ⇒ ((a +b)(b +c)(c +a)/a b c) ≥ 8 a b c ⇒ √[3](a b c/(a +b)(b +c)(c +a)) ≤ (1/2) ∴ From Eq. (i) (a/b+ c)+(b/c+ a)+(c/a+ b) ≥ (3/2)

Q. If $a, b, c > 0$ , the minimum value of $\frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b}$ is

1

$\frac{3}{2}$

2

$\frac{5}{2}$

Q. If a,b,c>0, the minimum value of b+ca​+c+ab​+a+bc​ is

1

23​

2

25​

Using AM≥GM 3b+ca​+c+ab​+a+bc​​≥3(a+b)(b+c)(c+a)abc​​ ...(i) Again using AM≥GM 2a+b​≥ab​,2b+c​≥bc​,2c+a​≥ca​ ⇒abc(a+b)(b+c)(c+a)​≥8abc ⇒3(a+b)(b+c)(c+a)abc​​≤21​ ∴ From Eq. (i) b+ca​+c+ab​+a+bc​≥23​

Q. If $a, b, c > 0$ , the minimum value of $\frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b}$ is

$\frac{3}{2}$

$\frac{5}{2}$