Find the smallest and largest values of the function: y = 3x-x² by (-3,0)

1. Let’s find the first derivative of the given function:

y ‘= (3x – x ^ 2)’ = 3 – 2x.

2. Let us equate this derivative to zero and find the critical points:

3 – 2x = 0;

-2x = -3;

x = -3: (-2);

x = 3/2;

x = 1.5.

Point x = 1.5 does not belong to the specified segment.

3. Find the value of the function at the ends of the given segment [-3; 0]:

y (-3) = 3 * (-3) – (-3) ^ 2 = -9 – 9 = -18;

y (0) = 3 * 0 – 0 = 0.

The largest value of the function is at the point x = 0, the smallest value of the function is at the point x = -3.

Answer: fmax = 0, fmin = -18.