Find the smallest value in the expression (x-3) ^ 2-2. Find the Largest Meaning of the Expression 6x-4x ^ 2

1) We have the expression:

(x – 3) ^ 2 – 2.

As you can see, there is a difference between the square of the binomial and a positive number. The square of the binomial always takes on non-negative values, which means that the expression takes its minimum value when the square of the binomial is zero, that is, -2.

2) Let’s write the expression as a function:

y = 6 * x – 4 * x ^ 2;

Find the derivative of the function:

y ‘= 6 – 8 * x;

Let’s find the critical points of the function:

6 – 8 * x = 0;

8 * x = 6;

x = 3/4;

If x <3/4, then the function is increasing.

If x> 3/4, then the function is decreasing.

x = 3/4 is the maximum point of the function.

y (3/4) = 6 * 3/4 – 4 * 9/16 = 9/2 – 9/4 = 9/4.