What are the x-coordinates of the points of inflection for the graph of f(x)sin^2x on the closed interval [0,pi]?
To find the x-coordinates of the points of inflection for the graph of f(x)sin^2x on the interval [0, pi], we need to follow these steps:
1. Differentiate the function f(x)sin^2x with respect to x to find the first derivative.
2. Set the first derivative equal to zero and solve for x to find the critical points.
3. Determine the second derivative of the function.
4. Substitute the critical points into the second derivative and solve for x to determine the x-coordinates of any inflection points on the interval [0, pi].
Let's go through each step one by one:
Step 1: Differentiate the function f(x)sin^2x
To find the derivative, we can use the product rule. The derivative of f(x)sin^2x is f'(x)sin^2x + 2f(x)sinx*cosx.
Step 2: Set the first derivative equal to zero and solve for x
Set f'(x)sin^2x + 2f(x)sinx*cosx = 0. Solve this equation for x to find the critical points.
Step 3: Determine the second derivative of the function
Differentiate the first derivative, f'(x)sin^2x + 2f(x)sinx*cosx, to find the second derivative, denoted as f''(x).
Step 4: Substitute the critical points into the second derivative and solve for x
Substitute the critical points obtained in Step 2 into the second derivative, f''(x). Solve this equation for x to determine the x-coordinates of any inflection points on the interval [0, pi].
By following these steps, you should be able to find the x-coordinates of the points of inflection for the graph of f(x)sin^2x on the closed interval [0, pi].