Cubic Max

https://www.desmos.com/calculator/vtfsl79yw6

At the right is the graph of the polynomial

It’s clear from the diagram that  is positive for x between 0 and 2.

The problem is to find the value of x (let that be ) at which the function  attains its maximum value on the interval [0, 2].

 Now this would be a standard problem in Grade 12 Calculus. The challenge here is to solve it without calculus.

This is a grade 12 Advanced Functions problem. There are a number of ways to tackle it, but students who had never seen anything like it would need help getting started. That is, they would need an idea. 

Here’s the idea. Drop the cubic curve down until the max point on the curve sits right on the x-axis. What we get is the graph of a new function:

where b of course is . What can you say about its factorization?