To find the exact location of the relative and absolute extrema of the function $f(x) = 3x^2 - 12x + 3$ on the interval $[0, 3]$, we can follow these steps:

Step 1: Find the first derivative of the function

The first derivative, $f'(x)$, will help us identify critical points where the slope of the function is zero or undefined.

$f'(x) = \frac{d}{dx} (3x^2 - 12x + 3) = 6x - 12$

Step 2: Set the first derivative equal to zero

To find critical points, set $f'(x) = 0$:

$6x - 12 = 0$

Solve for $x$:

$x = 2$

Thus, $x = 2$ is a critical point.

Step 3: Evaluate the function at the endpoints of the domain and at the critical point

The function is defined on the closed interval $[0, 3]$, so we need to evaluate $f(x)$ at both endpoints of the interval as well as at the critical point $x = 2$.

• $f(0) = 3(0)^2 - 12(0) + 3 = 3$
• $f(2) = 3(2)^2 - 12(2) + 3 = 12 - 24 + 3 = -9$
• $f(3) = 3(3)^2 - 12(3) + 3 = 27 - 36 + 3 = -6$

Step 4: Determine the relative and absolute extrema

Now, we compare the function values at the critical point and the endpoints:

• $f(0) = 3$

• $f(2) = -9$

• $f(3) = -6$

• The absolute maximum value is $3$, which occurs at $x = 0$.

• The absolute minimum value is $-9$, which occurs at $x = 2$.

• The relative maximum is not present because the function is continuously decreasing after the critical point $x = 2$.

Final Answer

1. $f(x)$ has an absolute maximum at $(0, 3)$.
2. $f(x)$ has a relative minimum at $(2, -9)$.
3. $f(x)$ has an absolute minimum at $(2, -9)$.

Would you like more details or have any further questions on this? Here are some related topics to explore:

1. How to identify critical points of a function.
2. The first and second derivative tests for extrema.
3. How to check for concavity and inflection points.
4. Understanding the difference between relative and absolute extrema.
5. The concept of a function's behavior on a closed interval.

Tip: Always check the endpoints of a function on a closed interval because the absolute extrema could occur there, even if there are critical points inside the interval.