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.

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

If x > -1, y” > 0,

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

So we are ready to sketch the curve:

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)