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)=(sec^−1 x)/x on [1,∞)

Verified step by step guidance

1. Step 1: Understand the function f(x) = \frac{\sec^{-1}(x)}{x} and its domain [1, \infty). The function involves the inverse secant function, which is defined for x \geq 1.
2. Step 2: Find the derivative f'(x) using the quotient rule. The quotient rule states that if you have a function h(x) = \frac{u(x)}{v(x)}, then h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}. Here, u(x) = \sec^{-1}(x) and v(x) = x.
3. Step 3: Calculate the derivatives u'(x) and v'(x). For u(x) = \sec^{-1}(x), use the derivative formula \frac{d}{dx}[\sec^{-1}(x)] = \frac{1}{|x|\sqrt{x^2 - 1}}. For v(x) = x, the derivative v'(x) = 1.
4. Step 4: Substitute u'(x), u(x), v'(x), and v(x) into the quotient rule formula to find f'(x). Simplify the expression to get the derivative in a manageable form.
5. Step 5: Set f'(x) = 0 to find the zeros of the derivative. These zeros correspond to the x-values where the original function f(x) has horizontal tangent lines. Verify these points by checking the graph of f(x) and f'(x) using technology.