Let
\[f(x) = (\arccos x)^3 + (\arcsin x)^3.\]Find the range of $f(x).$  All functions are in radians.
Question:
Let
\[f(x) = (\arccos x)^3 + (\arcsin x)^3.\]Find the range of $f(x).$  All functions are in radians.
Answer:
Let's think step by step.

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\left[ \frac{\pi^3}{32}, \frac{7 \pi^3}{8} \right]
Let
\[f(x) = (\arccos x)^3 + (\arcsin x)^3.\]Find the range of $f(x).$  All functions are in radians.
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