Find the smallest value of the function y = 0.5 ^ x in the interval [-1; 4]

Find the smallest value of the function y = 0.5 ^ x on the interval [-1; 4].

1) First, we find the derivative of the function y = 0.5 ^ x.

y ‘= (0.5 ^ x)’ = 0.5 ^ x * ln 0.5;

2) Let us equate the derivative of the function to 0 And find its roots.

0.5 ^ x * ln 0.5 = 0;

0.5 ^ x = 0;

No roots.

3) Find the values of the function y = 0.5 ^ x at the points x = -1 and x = 4.

y (-1) = 0.5 ^ (- 1) = 1 / 0.5 = 1 / (1/2) = 1/1 * 2/1 = 2/1 = 2;

y (4) = 0.5 ^ (4) = 0.5 * 0.5 * 0.5 * 0.5 = 0.25 * 0.25 = 0.0625.

Hence we find that the smallest value of the function y = 0.5 ^ x is 0.0625.

Answer: y min = 0.0625.