Final answer:
To find the maximum and minimum of xsinx in the given interval, we need to find the critical points and analyze their nature using the second derivative.
Step-by-step explanation:
To find the maximum and minimum of xsinx in the interval -π/2 to 3π/2, we need to find the critical points of the function and determine whether they are local maximum or minimum points.
The critical points occur when the derivative of xsinx equals zero or is undefined. Taking the derivative of xsinx, we get (1)(sinx) + (x)(cosx) = sinx + xcosx = 0.
Solving this equation, we can find the critical points. By analyzing the sign of the second derivative of the function at these points, we can determine whether they are local maximum or minimum points.