On which segment does the function y = 2 ^ x take the largest value equal to 32 and the smallest equal to 1/2.

The solution of the problem.

1. Let us investigate the function y (x) = 2 ^ x for critical points and monotonicity. For this we find the derivative of the function y (x).

y ‘(x) = 2 ^ x * ln 2.

2 ^ x> 0 for any x. ln 2> 0. Consequently, the derivative of the function y (x) exists and is greater than zero for any x. Accordingly, the function y (x) has no critical points and increases over the entire numerical interval.

2. It follows from the properties of the function y (x) that the function takes the greatest value at the extreme right point of the segment, and the smallest – at the extreme left point of the segment.

3. Determine the extreme points of the segment.

2 ^ x = 32;

x = 5 is the rightmost point of the line segment.

2 ^ x = 1/2;

x = -1 is the leftmost point of the line segment.

Answer. The largest value equal to 32 and the smallest value equal to 1/2, the function y = 2 ^ x takes on the segment [-1; 5].