Given the function $ f(x) = x^4 - 30x^2 $, determine all intervals on which $ f $ is increasing.

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To find the intervals where the function $ f(x) = x^4 - 30x^2 $ is increasing, we need to find the derivative $ f'(x) $ and determine where it is positive.

Find the derivative: $ f'(x) = 4x^3 - 60x $.
Set the derivative equal to zero to find critical points: $ 4x^3 - 60x = 0 $.
Factor the equation: $ 4x(x^2 - 15) = 0 $.
Solve for $ x $: $ x = 0 $ or $ x = \pm\sqrt{15} $.
Test intervals around the critical points to determine where $ f'(x) > 0 $:
- For $ x < -\sqrt{15} $, choose $ x = -4 $: $ f'(-4) = 4(-4)^3 - 60(-4) = -256 + 240 = -16 $ (not increasing).
- For $ -\sqrt{15} < x < 0 $, choose $ x = -1 $: $ f'(-1) = 4(-1)^3 - 60(-1) = -4 + 60 = 56 $ (increasing).
- For $ 0 < x < \sqrt{15} $, choose $ x = 1 $: $ f'(1) = 4(1)^3 - 60(1) = 4 - 60 = -56 $ (not increasing).
- For $ x > \sqrt{15} $, choose $ x = 4 $: $ f'(4) = 4(4)^3 - 60(4) = 256 - 240 = 16 $ (increasing).

Therefore, the function $ f(x) $ is increasing on the intervals $(-\sqrt{15}, 0)$ and $(\sqrt{15}, \infty)$.

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Given the function \( f(x) = x^4 - 30x^2 \), determine all intervals on which \( f \) is increasing.

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