Use the graph of f(x)=x^2/(x^2-4) to determine on which of the following intervals Rolle's Theorem applies. A) [0, 3] B) [-3, 3] C) [-3/2, 3/2] D) [-2, 2] E) None of these
Roll's theorem applies to a continuous interval. So to answer the question, you will first need to look for discontinuities in the given rational function.
You will get the hint by searching for at what point(s) the denominator vanishes (i.e. becomes zero).
To determine on which of the intervals Rolle's Theorem applies, we need to check two conditions:
1. The function must be continuous on the interval.
2. The function must be differentiable on the open interval.
Let's analyze the graph of f(x) = x^2/(x^2 - 4) to determine the validity of these conditions.
First, we need to check if the function is continuous on the given intervals. Looking at the graph, we can see that there are no vertical asymptotes or holes within the interval [-2, 2], so the function is continuous on that interval.
Next, we need to check the differentiability of the function on the open interval (-2, 2). For a function to be differentiable at a point, it must be continuous at that point, and the derivative must exist. Since the function is continuous on the interval [-2, 2], we should check if the derivative exists for all x-values within the open interval (-2, 2).
Differentiating f(x) = x^2/(x^2 - 4) with respect to x gives us:
f'(x) = (2x(x^2 - 4) - x^2(2x))/(x^2 - 4)^2
= (2x^3 - 8x - 2x^3)/ (x^2 - 4)^2
= -8x/ (x^2 - 4)^2
Since the denominator (x^2 - 4)^2 is never equal to zero for any x within (-2, 2), the derivative f'(x) = -8x/ (x^2 - 4)^2 exists for all x-values within the open interval (-2, 2).
Based on our analysis, both conditions of Rolle's Theorem are satisfied for the interval (-2, 2). Therefore, Rolle's Theorem applies to the interval D) [-2, 2].