The function $x^5-5 x^4+5 x^3-10$ has a maximum, when $x$ is equal to

Critical points of $f(x) = x^5 - 5x^4 + 5x^3 - 10$ are found where the first derivative vanishes.

The derivative is $f'(x) = 5x^4 - 20x^3 + 15x^2$, which factors as $5x^2(x-1)(x-3)$.
Setting $f'(x) = 0$ yields critical values at $x = 0$, $x = 1$, and $x = 3$.

The second derivative is $f''(x) = 20x^3 - 60x^2 + 30x$.
At $x = 1$, $f''(1) = -10 < 0$, indicating a local maximum.
At $x = 3$, $f''(3) = 90 > 0$, indicating a local minimum.
At $x = 0$, $f'(x)$ does not change sign, identifying it as an inflection point.

The maximum occurs at $x = 1$.