Sketch the graph of y=4(x^{2 }- x^{4}), locating the stationary points and giving their coordinates.
Syakirin,
I'm sure you know how to plug in values for x to sketch the graph, so I am just going to address the issues of the stationary points. Stationary points occur where the slope of the graph is zero (local minima and maxima).
To find these we can set the derivative of the equation dy/dx = 0 and solve for x.
so: d/dx [4(x² - x^{4}) = 4(2·x - 4·x³) = 0
We can factor out 4 ⇒ (2·x - 4·x³) = 0 by dividing through by 4.
Now do the same to remove a factor of 2 to get:
x - 2·x³ = 0 and rewrite as x·(1- 2·x²) = 0
Solutions that make this equation equal 0 are x = 0 and x = ±(1/√2)
Using these 3 values in the original equation and solving for y will give you the coordinates of the stationary points.