Question:

For the function: f(x) = -5(2x + 1) (x - 2)^{2 }(x - 6)

I was able to answer the following about this function:

Dominant term is -10x^{4}

The graph starts low and ends low. The roots are -1/2, 2 (bounces off), and 6.

I was able to sketch the graph without a calculator, roughly, and could see that the domain is {x e R}. From the answers section of the textbook, I saw that the range is {y e R, y < or equal to 500}. So the highest y value for the graph is 500.

My concern is, how is it possible to determine the maximum value of y (500) without a calculator? I believe there is a method in calculus but my course is a grade12 Advanced Functions course for Ontario, Canada and calculus is not taught.

Please help. If a calculus method is all that is available (short of using a calculator) please explain.