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.

Key Concepts

Derivative and Critical Points

The derivative of a function, denoted as f', represents the rate of change of the function f. Critical points occur where the derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. In this context, finding the zeros of f' helps identify where the function f has horizontal tangent lines, which are essential for understanding the function's behavior.

Horizontal Tangent Lines

A horizontal tangent line occurs at points on the graph of a function where the slope is zero. This means that the derivative of the function at those points is equal to zero. In the given problem, verifying that the zeros of f' correspond to horizontal tangents involves checking that these points indicate where the function f does not increase or decrease, thus providing insights into its local behavior.

Graphing Functions

Graphing a function involves plotting its values on a coordinate system to visualize its behavior. For the function f(x) = (x−1) sin^−1 x, understanding its graph helps in identifying critical points and the nature of its tangents. By analyzing the graph of both f and its derivative f', one can visually confirm the relationship between the zeros of f' and the horizontal tangents of f.