Answer:
Step-by-step explanation:
Recall that we use the first derivative to discover where a function is increasing or decreasing; it's increasing where the first derivative is + and decreasing where the first derivative is -.
The derivative of y = x^4 - 8x^2 + 16 is dy/dx = 4x^3 - 16x, or 4x(x^2 - 4).
This can be factored further: 4x(x - 2)(x + 2).
Set this equal to zero and solve for the three critical values:
{-2, 0, 2}.
Set up a total of four intervals: (-∞, -2), (-2, 0), (0, 2), (2, ∞).
Now choose a test point within each interval: {-3, -1, 1, 3}.
Evaluate the first derivative, dy/dx, at each of these four test points. Rule: if the first derivative is +, we know the function is increasing on that interval; if -, the function is decreasing.
At x = -3, dy/dx = 4x(x - 2)(x + 2) becomes (-)(-)(-), which is -, so we know that the function is decreasing on interval (∞, -2).
At x = -1, dy/dx is (-)(-)(+), which is +, so we know that the function is increasing on (-2, 0).
At x = 1, dy/dx is (+)(-)(+), which is -, so we know that the function is decreasing on (0, 2).
Finally: at x = 3, dy/dx is (+)(+)(+), so we know that the function is increasing on (2, ∞ ).
