A simple maximum

Calculus Level 2

Let $x$ be a positive real number.

If $f(x) = \dfrac{ 125x}{(15 +x)^{2}}$ , what value of $x$ maximizes $f(x)$ ?

2 solutions

Khang Nguyen Thanh
Aug 24, 2015

Using AM-GM inequality, we get:

$f(x)=\dfrac{125x}{(15+x)^2}\le\dfrac{125x}{\left(2\sqrt{15x}\right)^2}=\dfrac{25}{12}$ .

So $\max f(x)=\dfrac{25}{12}$ when $x=\boxed{15}$ .

The first, you must show the general formula(IM-GM) for the complete answer. Great!

Muhammad Aha - 5 years, 9 months ago

Denton Young
Aug 24, 2015

Take the derivative: $f'(x)$ has only one zero, at x = 15, indicating a maximum

You need to show by the second derivative test that it is a maximum value.

Pi Han Goh - 5 years, 9 months ago

Or you could graph it and see that it's a maximum, not a minimum. But yes, the second derivative test also shows that.

Denton Young - 5 years, 9 months ago