How many horizontal tangent lines exit to the curve f(x) = x^cos(x)?
To find the number of horizontal tangent lines that exit the curve f(x) = x^cos(x), we need to determine where the slope of the curve is equal to zero.
A horizontal tangent line occurs when the derivative of a function is equal to zero. Therefore, we need to find the derivative of f(x) and then solve for x when the derivative is equal to zero.
Step 1: Calculate the derivative of f(x)
Differentiating f(x) = x^cos(x) requires the application of the product rule and the chain rule. We will use the notation f'(x) to represent the derivative of f(x).
f'(x) = (cos(x)) * x^(cos(x) - 1) + x^cos(x) * (-sin(x))
Step 2: Set the derivative equal to zero and solve for x
Setting f'(x) = 0 and solving for x will give us the x-coordinates where the slope of the curve is zero.
(cos(x)) * x^(cos(x) - 1) + x^cos(x) * (-sin(x)) = 0
We need to solve this equation for x. This step might involve simplification, factoring, or using numerical methods such as graphing or solving algebraically.
Step 3: Determine the number of solutions
Count the number of unique solutions (values of x) obtained in Step 2. Each unique value of x corresponds to a point where the curve has a horizontal tangent line.
Note: The complexity of solving the equation and finding the exact number of solutions will depend on the specific equation and may require further mathematical analysis.
Once the solutions are found, you can determine the number of horizontal tangent lines that exit from the curve f(x) = x^cos(x) by counting the number of unique x-values.