To determine the range of the function f(x) = x4 – 4x2, we can start by rewriting the function for clarity:
f(x) = x2(x2 – 4)
Next, we can factor the quadratic part of the function:
f(x) = x2(x – 2)(x + 2)
This function is a quartic polynomial, and to find the range, we must analyze its behavior as x approaches positive and negative infinity, as well as its critical points.
The function goes to infinity as x approaches either positive or negative infinity. To find the local extrema, we compute the derivative:
f'(x) = 4x3 – 8x
Setting this equal to zero to find critical points:
4x(x2 – 2) = 0
From this, we find:
x = 0, x = ±√2
Now we can evaluate the function at these critical points:
f(0) = 0
f(√2) = (√2)4 – 4(√2)2 = 4 – 8 = -4
f(-√2) = f(√2) = -4
The minimum value of the function is at y = -4, which occurs at x = ±√2. Since the function approaches infinity in both directions, the range of the function can be expressed as:
Range of f(x): [−4, ∞)