Identify intervals on which the function is increasing, decreasing, or constant. g (x) = 4 - (x - 6) ^2?

This is a parabola, so it's easier to look at the expression than to compute derivatives (and maybe you have not covered that topic yet or are not allowed to use them in this exercise).

For x decreasing.

At x = 6, it gets its maximum value.

For x>6, the function goes from 4 to - infinity-> decreasing.

Taking the derivative will give you the velocity at any time.

g (x) = 4 - (x-6) ^2

g (x) = 4 - (x^2-12x+36)

g (x) = 4-x^2+12x-36

g (x) = - x^2+12x-32

dg/dx=-2x+12

So g (x) will be increasing when dg/dx>0

-2x+12>0

-2x>-12

x<6

So g (x) is increasing on the interval (-oo, 6)

g (x) will be decreasing when dg/dx<0

-2x+12<0

-2x<-12

x>6

So g (x) will be decreasing on the interval (6, + oo)