Find the greatest and smallest values of the function: y = x ^ 5-x ^ 3 + x + 2 on the segment [-1; 1].

1. Find stationary points:

y = x ^ 5 – x ^ 3 + x + 2;
U ‘= 5x ^ 4 – 3x ^ 2 + 1;
5x ^ 4 – 3x ^ 2 + 1 = 0;
D = 3 ^ 2 – 4 * 5 = 9 – 20 = -11 <0.
The equation has no solutions, there are no stationary points.

2. The first coefficient is greater than zero, it means that the derivative is positive, and the function increases. For an increasing function, the smallest value on a given segment [-1; 1] will be on the left, and the greatest value at the right end:

y = x ^ 5 – x ^ 3 + x + 2;
min (y) = y (-1) = (-1) ^ 5 – (-1) ^ 3 – 1 + 2 = -1 + 1 – 1 + 2 = 1;
Max (y) = y (1) = 1 ^ 5 – 1 ^ 3 + 1 + 2 = 1 – 1 + 1 + 2 = 3.
Answer: 1 and 3.