y = x⁴ - 4x³ + 2

dy/dx = 4x³ - 12x²

When dy/dx = 0,

4x³ - 12x² = 0
x³ - 3x² = 0
(x²)(x - 3) = 0

x² = 0 or x = 3
x = 0

So y = 2 or y = 3⁴ - 4(3³) + 2 = -25

Your stationary points are (0,2) and (3,-25)

d²y/dx² = 12x² - 24x

sub x = 0, d²y/dx² = 0

Here you have to check if it's a point of inflexion/inflection .

When you check 0+ and 0-, dy/dx are negative. So it's a downward point of inflexion/inflection

(Note : check for values as close to 0 as possible. Make sure that there are no other stationary points within the range you picked.)

sub x = 3, d²y/dx² = 12(3²) - 24(3) = 36
(3,-25) is a minimum point

Kathy, even though this chapter has been talking about cubic and power 4 equations, we cannot assume that all questions will be as such. The curve can be any equation, including sinusoidal and other curves.

You will next learn the trigonometric and exponential differentiations before there will be more questions on maxima minima and rates of change.