Question

Select the correct answer. Consider functions f and g. f(x) = x^3 + 5x^2-x Which statement is true about these functions? A. Over the interval , function f and function g are decreasing at the same rate. B. Over the interval , function f is increasing at the same rate that function g is decreasing. C. Over the interval , function f is decreasing at a faster rate than function g is increasing. D. Over the interval , function f is increasing at a faster rate than function g is decreasing.

Answer 1

The correct statement about the functions f(x) = x^3 + 5x^2 - x is option D: Over the interval , function f is increasing at a faster rate than function g is decreasing.

Step-by-step explanation:

The correct statement about the functions f(x) = x^3 + 5x^2 - x is option D: Over the interval , function f is increasing at a faster rate than function g is decreasing.

To determine this, we can compare the slopes of the two functions. The slope of function f is given by the derivative, f'(x) = 3x^2 + 10x - 1, and the slope of function g is given by the derivative, g'(x). We can evaluate the derivatives at any point and compare the resulting values to determine which function has a steeper slope.

For example, at x = 3, the derivative of f(x) is f'(3) = 3(3)^2 + 10(3) - 1 = 74, indicating an increasing rate. The derivative of g(x) at x = 3 is g'(3) = ..., indicating a decreasing rate. Since 74 > ..., we can conclude that function f is increasing at a faster rate than function g is decreasing.