Find the absolute maximum and minimum values of the function over the indicated​ interval, and indicate the​ x-values at which they occur.

Question

Let f(x)=x³ーx²ーx+2, [-1,2]

1. Find the absolute maximum and minimum values of f(x)over the indicated interval.
2. Indicate the x-values at which they occur.

Answer 1

Find the critical points: df/dx = 0 for 3x²-2x-1 = 0 (3x +1)(x-1) = 0 x = -1/3, 1)

Now just check these values by plugging these and the end points, -1 and 2.

You could check to see if the critical points are maxima or minima using 2nd derivative, but this only shows that they are local. You still have to check f(x) for the end points of the interval - You may as well just evaluate f(x) at the 4 points.