I don't know what you mean by "I have the problem solved up until I need to use the unit circle to substitute." Asking "should I take the derivatives" makes it look like you haven't actually done anything yet.

Most of the work in this problem is in differentiating to find where any local max or min inside the interval might be. After than you determine the global max and min by evaluating the function itself at those points and at the two endpoints, -1/4 and 6. For that-use a calculator!