The least value of the function `f(x) = ax + (b)/(x) (x gt 0, a gt 0, b gt 0)`

Explanation

The least value of function f(x) = `ax+b/x` (where, `a gt 0, b gt0, x gt 0)` is `2sqrt(ab)`.
`therefore f^(')(x)= a-b/x^(2)` and `f^(')(x)=0`
`rArr a=b/x^(2)`
`rArr x^(2)=b/a rArr x = +- sqrt(b/a)`
Now, `f^('')(x)=-b. (-2)/x^(3)= +(2b)/(x^(3))`
At `x=sqrt(b/a)`, `f^('')(x) = +(2b)/(b/a)^(3//2)=(+2b.a^(3//2))/(b^(3//2))`
`=+2b^(-1//2).a^(3//2)=+2sqrt(a^(3)/b) gt 0` `[therefore a, b gt 0]`
`therefore` Least value of f(x), `f(sqrt(b.a))= a.sqrt(b/a) + b/(sqrt(b/a))`
`=a.a^(-1//2). b^(1//2)+b.b^(-1//2).a^(1//2)`
`=sqrt(ab) + sqrt(ab)=2sqrt(ab)`