Find the largest and smallest values of the function y = 2-2x-x squared on the segment [2; 3].

In order to find the largest and smallest values of the function y = 2 – 2x – x ^ 2 on the segment [2; 3], let’s first analyze the function graph.

The graph of the function is a parabola whose branches are directed downward. This means that the function takes the greatest value at the vertex of the parabola. We are looking for its coordinates.

x0 = -b / 2a = 2 / -2 = -1;

-1 does not belong to the specified segment.

Then we will look for the largest and smallest value at the ends of the segment [2; 3].

y (2) = 2 – 2 * 2 – 2 ^ 2 = 2 – 4 – 4 = 2 – 8 = -6;

y (3) = 2 – 2 * 3 – 3 ^ 2 = 2 – 6 – 9 = 2 – 15 = -13.

Answer: ymax = y (2) = -6;

ymin (3) = -13.