Question

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

Answer 1

b>0 is the least valuof the function bt am not sure

Answer 2

Given:

$f(x) = ax + \frac{b}{x}$

a>0 , b> 0 , x> 0

To Find:

Least value of the function f (x)

Solution:

To find the least value of a function we need to differentiate the function with respect to x to obtain critical values of x.

$\frac{df}{dx} = 0$ (to find the critical values of x )

$\frac{df}{dy} = a - \frac{b}{x^{2} }$

differentiation of x = 1

differentiation of 1/x = -1/$x^{2}$

$a - \frac{b}{x^{2} } = 0$

x =+ $\sqrt{\frac{a}{b} }$ $or -\sqrt{\frac{a}{b} }$

but x >0 (given)

f($\sqrt{\frac{a}{b} }$) = ax + b/x= $a\sqrt{\frac{a}{b} } + \frac{b}{\sqrt{\frac{a}{b} }}$

least value of function f(x) is equal to $a\sqrt{\frac{a}{b} } + \frac{b}{\sqrt{\frac{a}{b} }}$