What is the range for f(x)=x^6+kx^4-9x^2-27?

Substitute for to find that this function is equivalent to:



We take the derivative with respect to to find the local minima, which occur at the solutions to:



Using the quadratic formula gives:



By the Trivial Inequality, since , , so .

To determine whether this is a local minimum or a local maximum, we compute the second derivative, which is .  This is positive for all positive values of (a reasonable assumption, since things get really messy really quickly otherwise), so the function is concave up at these values.  Hence, this value is a local minimum.

Since it is the only local minimum, we conclude that it is the global minimum, since as goes to positive or negative infinity, the term dominates and makes the function become extremely large.

The arithmetic gets pretty messy from here (I used WolframAlpha to ensure that I didn't make any mistakes), but in short, the rest of the problem pretty much devolves to substituting this value for back in, which gives you as the minimum.

Now, to show that there is no maximum, note that .  Hence, the range of is .