Need help finding a maximum value of a function?

Question

Find the maximum of function

f(x) = 4x +9(pi^2)/x + sinx

(x in range of (0,+infinite))

You can see the question in normal format at http://img510.imageshack.us/img510/6685/97761513ba7.jpg

I tried to differentiate the function and got

f'(x) = 4 - 9(pi^2)/(x^2) + cosx

but I am unable to find any value of x so that f'(x) = 0 so I got stuck right there.

Any solution? I don't think this problem is as hard as it looks...

Answer 1

you are correct that f'(x) never equals zero.

Because of that x in the denominator of f(x), at 0 the function is undefined and approaches infinity from the right (negative infinity from the left, but we are only considering positive x).

The behavior and appearance of the graph is similar to that of 1/x.

Answer 2

The function has no maximum. Check out the limit of the function as x goes to infinity. Do it term-wise.

The limit of 4x as x goes to infinity is infinity.

The limit of 9pi^2/x as x goes to infinity is 0.

And the limit of sin x as x goes to infinity does not exist. But that doesn't really matter, since sin x is bounded between -1 and 1.

The important thing to notice is that the 4x term is unbounded above and the other two are bounded, hence f(x) is unbounded above.