Determine whether the quadratic function shown below has a minimum or maximum, then determine the minimum or maximum value of the function.
f(x) = -(x – 6)2 – 1
The
value is

Step-by-step explanation:

f(x) = -(x-6)² - 1 = -(x² - 2×6x + 6²) - 1 = -x² + 12x - 37

since x² has a much bigger progression than 12x for growing values of x (and the constant -37 plays no role at all), -x² and with it the whole expression goes to -infinity when x goes to + or - infinity.

so, depending on the directions from your teacher the minimum is either -infinity or does not exist (as infinity is not a value).

now the maximum is at the extreme point of the curve. that is at the point where the square function curve turns around.

the easiest way to find this is finding the x value, for which the first derivative of f(x) is 0.

f'(x) = -2x + 12

-2x + 12 = 0

12 = 2x

x = 6

so, the maximum for f(x) happens at the point x=6.

and what is f(6) ?

f(6) = -(6)² + 12×6 - 37 = -36 + 72 - 37 = 72 - 73 = -1

so, the maximum of f(x) = -1