Find the smallest value of the function y = -4 / x- x on the segment [-2.5; -1]

1. Find the first derivative of the function y = (-4 / x) – x:

y ‘= (4 / x ^ 2) – 1.

2. Let us equate this derivative to zero:

(4 / x ^ 2) – 1 = 0;

(4 – x ^ 2) / x ^ 2 = 0;

4 – x ^ 2 = 0;

x ^ 2 ≠ 0;

x ^ 2 = 4;

x ≠ 0;

x1 = 2;

x2 = -2.

3. Find the value of the function at these points and at the ends of the given segment [-2,5; -one]:

y (2) = (-4/2) – 2 = -2 – 2 = -4;

y (-2) = (-4 / (- 2)) + 2 = 2 + 2 = 4;

y (-2.5) = (-4 / (- 2.5)) + 2.5 = 1.6 + 2.5 = 4.1;

y (-1) = (-4 / (- 1)) + 1 = 4 + 1 = 5.

The smallest value of the function on the segment [-2.5; -1] is 4 at x = -2.

Answer: fmin = 4.