The minimum value of the function $f(x) = x^3 + (10-x)^3$ occurs at:

To find the minimum value of $f(x) = x^3 + (10-x)^3$, the derivative is needed.

The derivative of $x^3$ is $3x^2$

The derivative of $(10-x)^3$ is $3(10-x)^2 \times (-1)$ by the chain rule.

$f'(x) = 3x^2 - 3(10-x)^2$

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At the minimum point, the derivative equals zero:

$3x^2 - 3(10-x)^2 = 0$

$3[x^2 - (10-x)^2] = 0$

$x^2 - (10-x)^2 = 0$

$x^2 = (10-x)^2$

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When two values squared are equal, the values are equal or opposite:

$x = 10-x$

$x + x = 10$

$2x = 10$

$x = 5$

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The minimum value occurs at $x = 5$.

At this point, both terms $x^3$ and $(10-x)^3$ are equal, creating symmetry: $f(5) = 5^3 + 5^3 = 125 + 125 = 250$