For what value of x will the function f (x) = - 3 (x - 10) (x - 4) have a maximum value? Find the maximum value.

Hello!

To find the maximum value of the function f (x) = - 3 (x - 10) (x - 4), the easiest way is to find the vertex using the formula: x = - b/2a.

Firstly, we need to simplify f (x).

f (x) = - 3 (x - 10) (x - 4)

f (x) = - 3 (x² - 14x + 40)

f (x) = - 3x² + 42x + - 120

Since the equation f (x) is now simplified to standard form, we can find the vertex.

a = - 3, b = 42, and c = - 120

x = - (42) / 2 (-3) = - 42/-6 = 7

Then, we substitute 7 into the the function f (x) = - 3 (x - 10) (x - 4), or

f (x) = - 3x² + 42x + - 120, to find the y-value of the vertex.

f (x) = - 3 (7 - 10) (7 - 4)

f (x) = - 3 (-3) (4)

f (x) = 27

The vertex of f (x) is (7, 27).

Therefore, the maximum x-value for the function f (x) is 7.