Question

The function `y=2x^3-3ax^2 + 6` is decreasing only on the interval (0,3). Find `a`?

Answer 1

The answer is `a = 3`.

Explanation:

If the function is decreasing on the interval (0, 3), that means its derivative must be negative on this interval. So first let's calculate the derivative, using the power rule:

`(dy)/(dx) 2x^3 - 3ax^2 + 6 = 6x^2 - 6ax`

So we know that the derivative `6x^2 - 6ax` is negative on the interval `(0,3)` and positive everywhere else. That means this function must cross the `x` axis at 0 and at 3. Or in other words, when `x=0` or when `x=3`, the function must be equal to 0. We can use this information to solve backwards for `a`.

No matter what `a` is, when `x=0`, the function will be 0, because `6*0^2 - 6*a*0 = 0`. So that doesn't help us.

But we also know that when `x=3`, the function also has to be 0, so we can write:
`6 * 9 - 6 * 3a = 0`
`54 = 18 a`
`a = 3`

To double check this, we can plug in `a = 3` and graph the function `y = 2x^3 - 9x^2 + 6`:

graph{2x^3 - 9x^2 + 6 [-3, 5, -26.6, 13.4]}

And we see that it is decreasing on the interval (0, 3).