Question

Create one function that satisfies all the following conditions and explain your choices:

-The instantaneous rate of change at x=2 is zero

-The instantaneous rate of change at x=3 is negative

-The average rate of change on the interval [0,4] is zero

Answer 1

draw a parabola opening down (sheds water) with vertex at (2,anything positive) and has zeros at (0,0) and (4,0)
let's see about the zeros
y = a x^2 + b x + c
c = 0 for (0,0)
y = a x^2 + b x
0 = a (16) + b(4)
so
a = -b/4 (negative coef for a so it opens down)
y = -b/4 x^2 + b x
now put in any positive number for b, like four
y = -x^2 + 4 x
when x = 2, y = 4
when x = 0 or x = 4 , y = 0 so average change from 0 to 34 is 0
the parabola is headed down at x = 3