Consider the function $$ f(x)=-7 x^{2}-14 x-6 $$ on the interval $$ [-3,2] $$. Find the absolute extrema for the function on the given interval. Express your answer as an ordered pair $$ (x, f(x)) $$. Answer

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We start with the function
$$
f(x) = -7 x^2 - 14 x - 6
$$
defined on the interval $$[-3,2]$$.

1. **Find the critical point (vertex of the parabola):**

Since $$ f(x) $$ is a quadratic function and the coefficient of $$ x^2 $$ is negative (i.e., $$ a = -7 $$), the parabola opens downward. Therefore, the vertex is the point of maximum.

The $$ x $$-coordinate of the vertex is given by
   $$
x = -\frac{b}{2a}
$$
where $$ a = -7 $$ and $$ b = -14 $$. Substituting these values,
   $$
x = -\frac{-14}{2(-7)} = \frac{14}{-14} = -1.
   $$

2. **Evaluate $$ f(x) $$ at the critical point and the endpoints:**

- At $$ x = -1 $$:
     $$
     f(-1) = -7(-1)^2 - 14(-1) - 6 = -7 + 14 - 6 = 1.
     $$
     
   - At the left endpoint $$ x = -3 $$:
     $$
     f(-3) = -7(-3)^2 - 14(-3) - 6 = -7(9) + 42 - 6 = -63 + 42 - 6 = -27.
     $$
   
   - At the right endpoint $$ x = 2 $$:
     $$
     f(2) = -7(2)^2 - 14(2) - 6 = -7(4) - 28 - 6 = -28 - 28 - 6 = -62.
     $$

3. **Determine the absolute extrema:**

- **Absolute Maximum:** The highest value is $$ f(-1) = 1 $$ at the critical point $$ x = -1 $$. So, the absolute maximum is 
   $$
(-1, 1).
   $$
   
   - **Absolute Minimum:** The lowest value is $$ f(2) = -62 $$ at the endpoint $$ x = 2 $$. So, the absolute minimum is 
   $$
(2, -62).
   $$

Thus, the absolute extrema of the function on the interval $$[-3,2]$$ are:
- Absolute maximum at $$\boxed{(-1, 1)}$$,
- Absolute minimum at $$\boxed{(2, -62)}$$.
The function $$ f(x) = -7x^2 - 14x - 6 $$ on the interval $$[-3,2]$$ has: - **Absolute Maximum** at $$ (-1, 1) $$, - **Absolute Minimum** at $$ (2, -62) $$.

Consider the function on the interval . Find the absolute extrema for the function on the given interval. Express your answer as an
ordered pair .
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Consider the function on the interval . Find the absolute extrema for the function on the given interval. Express your answer as an ordered pair . Answer