Final answer:
To find where the function F(x) = sinx - cosx is increasing and decreasing, we need to find the critical points of the function. Critical points occur where the derivative of the function is equal to zero or does not exist. To determine the concavity of the function, we need to find the second derivative.
Step-by-step explanation:
To find where the function F(x) = sinx - cosx is increasing and decreasing, we need to find the critical points of the function. Critical points occur where the derivative of the function is equal to zero or does not exist. Taking the derivative of F(x) gives us F'(x) = cosx + sinx. Setting F'(x) = 0 and solving for x, we get x = π/4 and x = 5π/4. These are the x-coordinates of the critical points.
To determine where the function is increasing and decreasing, we can examine the sign of the derivative in each interval. Since F'(x) = cosx + sinx, we can use a sign chart to analyze the intervals. We have:
Interval (-π, π/4): Since cosx and sinx are both positive in this interval, F'(x) > 0, so F(x) is increasing.
Interval (π/4, 5π/4): Since cosx is positive and sinx is negative in this interval, F'(x) < 0, so F(x) is decreasing.
Interval (5π/4, π): Since cosx and sinx are both negative in this interval, F'(x) > 0, so F(x) is increasing.
Interval (π, 2π): Since cosx is negative and sinx is positive in this interval, F'(x) < 0, so F(x) is decreasing.
To determine the concavity of the function, we need to find the second derivative. Taking the derivative of F'(x) = cosx + sinx, we get F''(x) = -sinx + cosx. We can use a similar analysis to determine the concavity of the function in each interval:
Interval (-π, π/2): Since -sinx + cosx is positive in this interval, the function is concave up.
Interval (π/2, 3π/2):
Since
-sinx + cosx
is negative in this interval, the function is
concave down
.
Interval (3π/2, 2π): Since -sinx + cosx is positive in this interval, the function is concave up.
The x-coordinates of the inflection points are the values of x where the concavity changes. In this case, the inflection points occur at x = π/2 and x = 3π/2 on the interval [-π, π].