Find the largest and smallest value of the function y = x / 2 + x ^ 3 on the segment [0; 3].

1. Find the derivative of the function y (x). y ‘(x) = 1/2 + 3x ^ 2.
2. Let us examine the function y ‘(x). y ‘(x) = 1/2 + 3x ^ 2. The first term is greater than zero, the second term is greater than or equal to zero. y ‘(x) = 1/2 + 3x ^ 2> 0 for any value of x.
3. The function y (x) increases over the entire numerical interval, there are no extreme points.
4. y (0) = 0 is the smallest value of the function y (3) = 3/2 + 27 = 28.5 is the largest value of the function.
Answer. On the segment [0; 3], the smallest value of the function is 0, the largest value of the function is 28.5.