# Find the intervals on which f(x) = x^{4} − 8x^{2} 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) = x^{4} − 8x^{2 }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) = x^{4} − 8x^{2}.

Now, to determine the monotonicity of the function f(x) = x^{4} − 8x^{2}, we first differentiate f(x).

⇒ f(x) = x^{4} − 8x^{2}

⇒ f'(x) = 4x^{3} - 16x

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

⇒ 4x^{3} - 16x = 0

⇒ 4x(x^{2} - 4) = 0

⇒ 4x(x - 2)(x + 2) = 0 [using a^{2} - b^{2} = (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) = x^{4} − 8x^{2 }is increasing in the interval (-2, 0) ∪ (2, ∞), and decreasing in the interval (−∞, −2) ∪ (0, 2).

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