Given the function $$ f(x)=\int_{-1}^{x}\left(t^{4}-12 t^{3}\right) d t $$ determine all $$ x $$-values, if any, for which $$ f $$ has a relative maximum.

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To determine the $$ x $$-values for which the function $$ f(x) $$ has a relative maximum, we will follow these steps:

1. **Find the derivative $$ f'(x) $$** using the Fundamental Theorem of Calculus.
2. **Set the derivative equal to zero** to find critical points.
3. **Use the second derivative test** to determine if these critical points correspond to relative maxima.

### Step 1: Find the derivative $$ f'(x) $$

According to the Fundamental Theorem of Calculus, the derivative of the integral function is given by:

$$
f'(x) = \frac{d}{dx} \left( \int_{-1}^{x} (t^{4} - 12t^{3}) dt \right) = x^{4} - 12x^{3}
$$

### Step 2: Set the derivative equal to zero

Now, we will set $$ f'(x) $$ equal to zero to find the critical points:

$$
x^{4} - 12x^{3} = 0
$$

Factoring out $$ x^{3} $$:

$$
x^{3}(x - 12) = 0
$$

This gives us the critical points:

$$
x^{3} = 0 \quad \Rightarrow \quad x = 0
$$
$$
x - 12 = 0 \quad \Rightarrow \quad x = 12
$$

### Step 3: Use the second derivative test

Next, we need to find the second derivative $$ f''(x) $$:

$$
f''(x) = \frac{d}{dx}(f'(x)) = \frac{d}{dx}(x^{4} - 12x^{3}) = 4x^{3} - 36x^{2}
$$

Now we will evaluate $$ f''(x) $$ at the critical points $$ x = 0 $$ and $$ x = 12 $$:

1. **At $$ x = 0 $$**:

$$
f''(0) = 4(0)^{3} - 36(0)^{2} = 0
$$

Since $$ f''(0) = 0 $$, the second derivative test is inconclusive at this point.

2. **At $$ x = 12 $$**:

$$
f''(12) = 4(12)^{3} - 36(12)^{2}
$$

Calculating $$ f''(12) $$:

$$
= 4(1728) - 36(144) = 6912 - 5184 = 1728
$$

Since $$ f''(12) > 0 $$, this indicates that $$ f(x) $$ has a relative minimum at $$ x = 12 $$.

### Conclusion

The only critical point we found was $$ x = 0 $$, which was inconclusive, and $$ x = 12 $$, which is a relative minimum. Therefore, there are no $$ x $$-values for which $$ f $$ has a relative maximum.

Thus, the final answer is:

**There are no $$ x $$-values for which $$ f $$ has a relative maximum.**
There are no $$ x $$-values where $$ f $$ has a relative maximum.

Given the function \( f(x)=\int_{-1}^{x}\left(t^{4}-12 t^{3}\right) d t \) determine all \( x \)-values, if any, for which \( f \) has a relative maximum.

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