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.
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.