Find the largest value of the function y = -1.5x ^ 2 on the segment [-4; -2].

Let’s find the largest value of the function y = -1.5x ^ 2 on the segment [-4; -2], for this:

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

y ‘= (-1.5x ^ 2)’ = -1.5 * 2x = -3x.

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

-3x = 0;

x = 0.

Point x = 0 does not belong to the specified segment.

3. Find the value of the function at the ends of a given segment:

y (-4) = -1.5 * (-4) ^ 2 = -1.5 * 16 = -24;

y (-2) = -1.5 * (-2) ^ 2 = -1.5 * 4 = -6.

The largest value of the function on the segment [-4; -2] is at the point x = -2 and equals -6.

Answer: fmax = -6.