Find the intervals on which f(x) = x⁴ − 8x² is increasing and decreasing.

Derivatives have many applications. One of the major applications of derivatives is determining the monotonicity of a given function in a specific interval.

Answer: The function f(x) = x⁴ − 8x² is increasing in the interval (-2, 0) ∪ (2, ∞), and decreasing in the interval (−∞, −2) ∪ (0, 2).

Let's understand the solution in detail.

Explanation:

Given function: f(x) = x⁴ − 8x².

Now, to determine the monotonicity of the function f(x) = x⁴ − 8x², we first differentiate f(x).

⇒ f(x) = x⁴ − 8x²

⇒ f'(x) = 4x³ - 16x

Now, we equate f'(x) to zero.

⇒ 4x³ - 16x = 0

⇒ 4x(x² - 4) = 0

⇒ 4x(x - 2)(x + 2) = 0 [using a² - b² = (a - b)(a + b)]

Hence, we have the solutions x = 0, x = -2 and x = 2.

Now, we can observe that the value of f'(x) is positive in the range (-2, 0) ∪ (2, ∞). Hence it is increasing in that range.

Also, we can observe that the value of f'(x) is negative in the range (−∞, −2) ∪ (0, 2). Hence it is decreasing in that range.

Thus, the function f(x) = x⁴ − 8x² is increasing in the interval (-2, 0) ∪ (2, ∞), and decreasing in the interval (−∞, −2) ∪ (0, 2).