62​–​65. {Use of Tech} Graphing f and f'
c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.
f(x) = (x−1) sin^−1 x on [−1,1]

Verified step by step guidance

1. Step 1: Understand the problem. We need to verify that the zeros of the derivative of the function f(x) = (x-1) \sin^{-1}(x) correspond to points where the original function f(x) has a horizontal tangent line.
2. Step 2: Find the derivative f'(x). Use the product rule for differentiation, which states that if you have a function h(x) = u(x)v(x), then h'(x) = u'(x)v(x) + u(x)v'(x). Here, u(x) = (x-1) and v(x) = \sin^{-1}(x).
3. Step 3: Differentiate u(x) and v(x). The derivative of u(x) = (x-1) is u'(x) = 1. The derivative of v(x) = \sin^{-1}(x) is v'(x) = \frac{1}{\sqrt{1-x^2}}.
4. Step 4: Apply the product rule. Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula to find f'(x). This gives f'(x) = 1 \cdot \sin^{-1}(x) + (x-1) \cdot \frac{1}{\sqrt{1-x^2}}.
5. Step 5: Find the zeros of f'(x). Set f'(x) = 0 and solve for x. These x-values are where the derivative is zero, indicating potential horizontal tangent lines on the graph of f(x). Verify these points by checking the graph of f(x) to see if the tangent is indeed horizontal at these x-values.