Consider the function $$ f(x)=-4 x^{3}+12 x^{2}+96 x $$ on the interval $$ [-8,6] $$. Find the absolute extrema for the function on the given interval. Express your answer as an ordered pair $$ (x, f(x)) $$.

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1. Compute the derivative of the function:
   $$
   f(x) = -4x^3 + 12x^2 + 96x
   $$
   $$
   f'(x) = -12x^2 + 24x + 96
   $$

2. Set the derivative equal to zero to find the critical points:
   $$
   -12x^2 + 24x + 96 = 0
   $$
   Divide both sides by $$-12$$:
   $$
   x^2 - 2x - 8 = 0
   $$
   Factor the quadratic:
   $$
   (x - 4)(x + 2) = 0
   $$
   Thus, the critical points are:
   $$
   x = 4 \quad \text{and} \quad x = -2
   $$

3. Evaluate the function at the critical points and at the endpoints $$ x = -8 $$ and $$ x = 6 $$:

- At $$ x = -8 $$:
     $$
     f(-8) = -4(-8)^3 + 12(-8)^2 + 96(-8)
     $$
     $$
     f(-8) = -4(-512) + 12(64) - 768 = 2048 + 768 - 768 = 2048
     $$
   
   - At $$ x = -2 $$:
     $$
     f(-2) = -4(-2)^3 + 12(-2)^2 + 96(-2)
     $$
     $$
     f(-2) = -4(-8) + 12(4) - 192 = 32 + 48 - 192 = -112
     $$
   
   - At $$ x = 4 $$:
     $$
     f(4) = -4(4)^3 + 12(4)^2 + 96(4)
     $$
     $$
     f(4) = -4(64) + 12(16) + 384 = -256 + 192 + 384 = 320
     $$
   
   - At $$ x = 6 $$:
     $$
     f(6) = -4(6)^3 + 12(6)^2 + 96(6)
     $$
     $$
     f(6) = -4(216) + 12(36) + 576 = -864 + 432 + 576 = 144
     $$

4. Identify the absolute extrema:
   - The absolute maximum is $$ f(-8) = 2048 $$ at $$ x = -8 $$, so the ordered pair is:
     $$
     (-8, 2048)
     $$
   - The absolute minimum is $$ f(-2) = -112 $$ at $$ x = -2 $$, so the ordered pair is:
     $$
     (-2, -112)
     $$
The absolute maximum is at $$ (-8, 2048) $$ and the absolute minimum is at $$ (-2, -112) $$.

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 .