Question

Find the absolute minimum and absolute maximum of F on the given closed interval by using the candidates test and state where these values occur for the function f(x)=sinx-cosx [0, pi]

Answer 1

The absolute minimum of the function f(x) = sinx - cosx on the interval [0, π] is -1, occurring at x = 0, and the absolute maximum is 1, occurring at x = π.

This is found using the candidate's test by computing the derivative, setting it to zero to find critical points, and evaluating the function at these points and the endpoints.

Step-by-step explanation:

To find the absolute minimum and absolute maximum of the function f(x) = sinx - cosx on the closed interval [0, π], we use the candidate's test.

1. First, find the derivative of the function: f'(x) = cosx + sinx.
2. Set the derivative equal to zero to find critical points: cosx + sinx = 0.
3. Solve for x to find the critical points within the interval [0, π]: sinx = 0 or cosx = 0.
4. Check the value of f(x) at the critical points and the endpoints of the interval: f(0), f(π), and the values from the critical points.
5. Compare these values to determine the absolute minimum and maximum.

Computing the function values:

• f(0) = sin(0) - cos(0) = 0 - 1 = -1.
• f(π) = sin(π) - cos(π) = 0 - (-1) = 1.

For the critical points, sinx = -cosx implies tanx = -1, which within the interval [0, π] happens at x = π/4.

Evaluating at this point gives us f(π/4) = sin(π/4) - cos(π/4)

= √2/2 - √2/2

= 0.

Comparing the values, we see that the absolute minimum value is -1 at x = 0 and the absolute maximum is 1 at x = π.