Find the smallest value of the function x ^ 5 -5x + 8 on the segment [-2; 3]

1. Stationary points of the function:

f (x) = x ^ 5 – 5x + 8;
f ‘(x) = 5x ^ 4 – 5 = 5 (x ^ 4 – 1);
5 (x ^ 4 – 1) = 0;
x ^ 4 – 1 = 0;
x ^ 4 = 1;
x = ± 1.
2. Both stationary points belong to the specified interval [-2; 3]. To find the smallest value of a function on this segment, we calculate its value at four points:

f (x) = x ^ 5 – 5x + 8;
f (-2) = (-2) ^ 5 – 5 * (-2) + 8 = -32 + 10 + 8 = -14;
f (-1) = (-1) ^ 5 – 5 * (-1) + 8 = -1 + 5 + 8 = 12;
f (1) = 1 ^ 5 – 5 * 1 + 8 = 1 – 5 + 8 = 4;
f (3) = 3 ^ 5 – 5 * 3 + 8 = 243 – 15 + 8 = 236.
fmin = -14.
Answer: -14.