Final answer:
To analyze if the function f(x)=1.2x^4-5.7x^2+4.09 is increasing or decreasing, we consider its first derivative. Examining this will reveal the slope at any given point and thus the function's behavior, whether it is increasing or decreasing.
Step-by-step explanation:
The question asks about the behavior of the function f(x)=1.2x4-5.7x2+4.09, specifically whether it's increasing or decreasing, and for an analysis of any patterns in its behavior. To determine if the function is increasing or decreasing at a certain point, we need to examine its first derivative, f'(x). The first derivative will tell us the slope of the function at any given x-value. If f'(x) is positive, the function is increasing at that x-value; if f'(x) is negative, the function is decreasing. On analyzing the first derivative of the given function, we can identify intervals on which the function is increasing or decreasing.
In general, the degree of the polynomial indicates it will have at most three turning points and the leading coefficient being positive means the end behavior will resemble that of x4, that is, up on both ends of the x-axis. Knowing these characteristics can help us sketch a rough graph of the function's behavior.