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)=(x²−1)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^2 - 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, since f(x) is a product of two functions: u(x) = x^2 - 1 and v(x) = sin^{-1}(x). The product rule states that (uv)' = u'v + uv'.
3. Step 3: Differentiate u(x) and v(x) separately. For u(x) = x^2 - 1, the derivative u'(x) = 2x. For v(x) = sin^{-1}(x), the derivative v'(x) = 1/√(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) = (x^2 - 1)(1/√(1-x^2)) + (2x)(sin^{-1}(x)).
5. Step 5: Set f'(x) = 0 to find the zeros of the derivative. Solve the equation (x^2 - 1)(1/√(1-x^2)) + (2x)(sin^{-1}(x)) = 0 to find the x-values where f'(x) is zero. These x-values correspond to points where f(x) has a horizontal tangent line.

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 locations for horizontal tangent lines. In this context, finding the zeros of f' helps identify where the slope of f is zero, which corresponds to horizontal tangents.

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. Understanding this concept is crucial for verifying the relationship between the zeros of the derivative and the behavior of the original function.

Graphing Functions

Graphing a function involves plotting its values on a coordinate plane to visualize its behavior. By graphing both f(x) and its derivative f'(x), one can observe how the zeros of f' correspond to points where f has horizontal tangents. This visual representation aids in understanding the relationship between a function and its derivative.