Question

For which values of `a` the function `f(x) = a^2x^3+3x` is growing on all the domain?

Answer 1

Any values of `a` will suffice

Explanation:

We want to determine the value of `a` so that `f(x)` is increasing on all `x`. To do this, we're going to have to find the first derivative.

`f'(x) = a^2(3x^2) + 3`

`f'(x) = 3a^2x^2 +3`

Therefore we need `3a^2x^2 + 3 > 0`

We solve this as an equation and select test points.

`0 = 3a^2x^2 +3`

`-3 = 3a^2x^2`

`x = sqrt(-1/a^2)`

We see that there is no value of `a` that satisfies, because the number under the square root will always be negative. If we test any values of `a` and `x` in the derivative you se it'll be positive.

Thus, all values of `a` will work.

Hopefully this helps!