The maximum value of f(x) = 6 + 6x - 4x^2 has to be determined without the use of calculus.

One way of doing this would be to graph the function and look at what its highest value is.

The maximum value of the function is slightly greater than 8.

Another way, that allows us to determine the exact value is as follows.

6 + 6x - 4x^2

= 6 + 6x - 4x^2 - 9/4 + 9/4

= 6 + 9/4 - (2x - 3/2)^2

= 33/4 - (2x - 3/2)^2

The minimum value of (2x - 3/2)^2 is 0. As the second term is the square of (2x - 3/2) it cannot take on a negative value.

This gives the maximum value of f(x) = 6 + 6x - 4x^2 as 33/4 = 8.25

**The maximum value of f(x) = 6 + 6x - 4x^2 is 8.25**

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