1. x-intercepts

`y=x^4-6x^2 `

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

`=x^2(x+sqrt6)(x-sqrt6)`

`=0`

when

`x = 0`, `x=-sqrt(6)` and `x=sqrt(6)`

2. y-intercepts:

When `x = 0`, `y = 0`.

3. maxima and minima?

`(dy)/(dx)=4x^3-12x `

`=4x(x^2-3) `

`=4x(x+sqrt3)(x-sqrt3)`

`=0`

Now `(dy)/(dx)=0` when `x = 0` or `x=-sqrt(3)` and `x=sqrt(3)`

So we have max or min at `(0,0)` and `(-sqrt(3),-9)` and `(sqrt(3),-9)`.

4. Second derivative:

`(d^2y)/(dx^2)=12x^2-12`

Now `y” > 0` for `x = -sqrt3` so `(-sqrt3, -9)` is a local MIN

Now `y” < 0` for `x = 0` so `(0, 0)` is a local MAX

Now `y” > 0` for `x = sqrt3` so `(sqrt3, -9)` is a local MIN

5. Points of inflection:

We now use the second derivative to find points of inflection:

`(d^2y)/(dx^2)=12x^2-12`

`=12(x+1)(x-1)`

`=0`

when `x = -1` or `x = 1`

If `x < -1`, `y” > 0`, and for `-1 < x < 1`, we have `y” < 0`.

The sign of `y”` has changed, so `(-1, -5)` is a point of inflection.

If `x > 1`, `y” > 0`,

The sign of `y”` has changed, so `(1, -5)` is a point of inflection.

So we are ready to sketch the curve:

Open image in a new page

Graph of `y=x^4-6x^2`.

The following points are indicated with dots:

`x`-intercepts `(-sqrt(6),0)` and `(sqrt(6),0)` (green dots)

Local maximum, `x`-intercept and `y`-intercept `(0,0)` (green dot)

Points of inflection `(-1,-5)` and `(1,-5)` ("plus" signs)

Local minima `(-sqrt(3),-9)` and `(sqrt(3),-9)` (magenta dots)