Question

On which interval does the function g(x)=4x–3x^2 have an average rate of change equal to 16?

Answer 1

The bot is wrong again!



There would be an infinite number of such intervals.

e.g. let the left point be (0,0), then we want the intersection point of
the line y = 16x with the parabola .....
16x = 4x - 3x^2
3x^2 + 12x = 0
3x(x + 4) = 0
x = 0 or x = -4

if x = 0, y = 0 , our starting point
if x = -4, y = -64 , the other point in our required domain.
slope from (-4,-64) to (0,0) = (0+64)/(0+4) = 16
so an interval where the average slope is 16 would be [-4,0]

We could proceed this way for any starting point on the parabola,
find the equation of the line with slope 16 going through that starting
point and find the other intersection point.

Interesting that the interval of [-2,2] given by the bot is wrong.
Of the billions and billions and billions or possibilities, the bot picked
a wrong one!!!!

Proof:
f(-2) = -8 - 12 = -20
f(2) = 8 - 12 = -4
average slope = (-4+20)/(2+2) = 4 , not 16 as required.

PS, did this mostly for fun and my own satisfaction, since I know that
the bot doesn't "care", and at most only about 5 real students will actually
look at this.

Answer 2

The interval is [-2,2].