Final answer:
To determine the end behavior of the function f(x), we should evaluate f(x) as x approaches positive and negative infinity (option A). The highest power term, 3x⁴, dictates that as x approaches infinity in either direction, the function will approach positive infinity.
Step-by-step explanation:
To determine the end behavior of the function f(x)=3x⁴ − 2x³ + 5x² − 7, we should consider how the function behaves as x approaches positive infinity and negative infinity, which aligns with option A. Since this function is a polynomial, its end behavior is primarily determined by the term with the highest power, in this case, 3x⁴. As x approaches positive infinity or negative infinity, the x⁴ term will dominate the value of the function, and because it is a positive coefficient, the function will tend to positive infinity in both directions.
The derivative of f(x), option B, gives us information about the slope of the function but does not directly indicate the end behavior. The integral of f(x), option C, would instead measure the area under the curve, which is also not directly related to the end behavior. Lastly, option D, calculating the square root of f(x), is not standard for evaluating end behavior and, in many cases, is not practical due to the polynomial having both positive and negative values.