Question

The least value of the function `f(x) = ax + b/x (a > 0, b > 0, x > 0)` is ______.

Answer 1

The least value of the function `f(x) = ax + b/x (a > 0, b > 0, x > 0)` is `underlinebb(2sqrt(ab))`.

Explanation:

Here, `f(x) = ax + b/x`

For the least value

f'(x) = `a - b/x^2 = 0`

⇒ `a = b/x^2`

⇒ `x = sqrt(b/a)`

Also, f"(x) = `+ (2b)/x^3`

Then, f"`(sqrt(b/a)) = (2b)/(b/a)^(3/2) > 0`

Least value of the function is

`f(sqrt(b/a)) = asqrt((b/a)) + b/sqrt((b/a))`

= `sqrt(ab) + sqrt(ab)`

= `2sqrt(ab)`

∴ `f(sqrt(b/a)) = 2sqrt(ab)`