Question

How do you test `f(x)=8 x^4−9 x^3 +9` for concavity and inflection points?

Answer 1

Explanation:

To test for the concavity and inflection points you need to equate the second order derivative with zero.

Keeping in mind:

• `d/dxx^n=nx^(n-1)`

• `d/dxc=0`

We proceed:

`f(x)=8x^4-9x^3+9`

`=>f'(x)=32x^3-27x^2`

`=>f''(x)=96x^2-54x`

`=6(16x^2-9x)`

Now,

`f''(x)=0`

`=>6(16x^2-9x)=0`

`=>x(16x-9)=0`

`=>color(red)(x=0,x=9/(16))` are the inflection points. Inflection points are those points where the curve changes its concavity if any.

graph{x(16x -9) [-5, 5, -5, 5]}

Sign Chart: See image.

Now, to determine the opening of the concavity.

• Put any value less than `0` in `f''(x)`.

`f''(x)` comes out to be positive. `(+)`.

• Put any value between `0` to `9/(16)`.

`f''(x)` comes out to be negative. `(-)`.

• Put any value greater than `9/(16)`.

`f''(x)` comes out to be positive. `(+)`.

Negative sign indicates that the curve will open downwards. And positive sign indicates it'll open up.

Thus, `(-oo,0)∪(9/(16),oo)` our concavity is upwards.

And,

`(0,9/(16))` our concavity is downwards.

Hope this helps. :)