Did you know?
Did you know that by analyzing the function T(z) = sin^2(z/4) over the interval -3 ≤ z ≤ 8, we can determine various properties of the function?
To identify where T(z) is increasing, we need to find where the derivative is positive. By taking the derivative of T(z) and finding its critical points, we can determine these intervals.
Similarly, to find where T(z) is decreasing, we need to identify the intervals where the derivative is negative.
To determine where T(z) is concave up or down, we need to find the second derivative of T(z). The intervals where the second derivative is positive indicate concave up, while negative intervals indicate concave down.
Lastly, we can identify the absolute minimum of T(z) by locating the lowest point on the graph. This minimum may occur at an endpoint or a critical point within the interval.
Additionally, there may be a relative maximum (not an absolute maximum) for T(z), which is the highest point on the graph, excluding the endpoints. This maximum can be found by analyzing the critical points in the interval.
By analyzing these properties, we gain a deeper understanding of the behavior and characteristics of T(z) = sin^2(z/4).
