Let f(x)=(x+3)2(x−2)3,x∈[−4,4]. If M and m are the maximum and minimum values of f, respectively in [−4,4], then the value of M−m is :

To find the maximum and minimum values of the function f(x) = (x+3)^2 * (x-2)^3 on the interval [-4,4], we first need to find the critical points of the function. The critical points are where the derivative of the function is zero or undefined.

Step 1: Find the derivative of the function. f'(x) = 2(x+3)(x-2)^3 + 3(x+3)^2(x-2)^2

Step 2: Set the derivative equal to zero and solve for x. 0 = 2(x+3)(x-2)^3 + 3(x+3)^2(x-2)^2 This gives us the critical points x = -3, 2.

Step 3: Evaluate the function at the critical points and the endpoints of the interval. f(-4) = (-4+3)^2 * (-4-2)^3 = 1 * -216 = -216 f(-3) = (-3+3)^2 * (-3-2)^3 = 0 f(2) = (2+3)^2 * (2-2)^3 = 25 * 0 = 0 f(4) = (4+3)^2 * (4-2)^3 = 49 * 8 = 392

Step 4: The maximum value M is the largest of these values, and the minimum value m is the smallest. So, M = 392 and m = -216.

Step 5: Subtract the minimum value from the maximum value. M - m = 392 - (-216) = 608.