Final answer:
To find the function with the greatest rate of change on the interval from x=π to x=3π/2, we can compare the slopes of the functions at those points. After evaluating the slopes, we can determine that the function with the greatest rate of change is g(x). Hence the correct answer is option B
Step-by-step explanation:
To determine which function has the greatest rate of change on the interval from x=π to x=3π/2, we need to compare the slopes of the functions at those points.
The rate of change of a function is represented by its slope. The steeper the slope, the greater the rate of change.
In this case, we can evaluate the slopes of the functions at x=π and x=3π/2 to compare.
Let's label the functions as f(x), g(x), and h(x) for convenience.
To find the slope of a function, we can take the derivative with respect to x.
Let's find the derivatives of the functions:
f'(x) = 2x
g'(x) = -3x
h'(x) = 1
Now, let's evaluate the slopes at x=π and x=3π/2:
f'(π) = 2π ≈ 6.283
f'(3π/2) = 2(3π/2) = 3π ≈ 9.425
g'(π) = -3π ≈ -9.425
g'(3π/2) = -3(3π/2) = -9π/2 ≈ -14.137
h'(π) = 1
h'(3π/2) = 1
Comparing the slopes at these points, we can see that the function with the greatest rate of change is g(x).
Therefore, the answer is B) g(x).