If `ax^(2)+b/x ge c, AA x gt0`, where `agt0,bgt0` then

Options

A. `27ab^(2)ge4c^(3)`
B. `27ab^(3) le 4c^(3)`
C. `ab^(2) gec^(3)`
D. `ab^(3)lec^(3)`

Answer

A

Explanation

Let `f(x)=ax^(2)+b/x-c`
`impliesf'(x)=2 ax -b/(x^(2))`
If `f'(x)=0implies2ax=b/(x^(2))=0impliesx=(b/(2a))^(1//3)gt0`
But `f(x)ge0AAxgt0`
`:.f(b/(2a))^(1//3)ge0`
`impliesa.(b/(2a))^(2//3)+b((2a)/b)^(1//3)gec`
`implies a .b.(2a)+bgec.(b/(2a))^(1//3)`
`implies 3bge2c(b/(2a))^(1//3)`
`implies27b^(3)ge8c^(3)(b/(2a))`
`implies27ab^(2)ge4c^(3)`