Polynomials

We are asked about the nature of the point for the function: To determine whether it is a maximum, minimum, or neither, we analyze the derivative.

Step 1: First Derivative using Product Rule
Let: Then:

Differentiate: So:

Factor common terms:

Simplify the bracket: So:

Step 2: Check critical point at
Set . This happens when:

So is a critical point. Now determine its nature.

Step 3: Use the first derivative test near

• Choose , say : , , since even power
• ,
• So just before 2
• Now take , say : , all other terms also positive
• So after 2 as well

Conclusion: does not change sign at , and remains positive on both sides. So the graph is increasing before and after \Rightarrow it is neither a maximum nor a minimum

Answer: