Let $$ f(x)=3 x^{2}-12 x+9 $$. (a) Find the critical point $$ c $$ of $$ f $$ and compute $$ f(c) $$. (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list if needed. Enter NCP if the function has no critical points. Enter DNE if there are no maximum or minimum.) $$ c=\square $$ $$ f(c)=\square $$ (b) Find extreme values of $$ f $$ on $$ [0,5] $$. Maximum $$ =\square $$ Minimum $$ =\square $$ (c) Find extreme values of $$ f $$ on $$ [-8,1] $$. Maximum $$ =\square $$ Minimum $$ =\square $$

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$$ \textbf{(a)} $$

1. Compute the derivative:
$$
f'(x)=\frac{d}{dx}\left(3x^2-12x+9\right)=6x-12.
$$

2. Set the derivative to zero to find the critical point:
$$
6x-12=0 \quad\Longrightarrow\quad x=2.
$$
Thus, the critical point is $$ c=2 $$.

3. Evaluate $$ f $$ at $$ x=2 $$:
$$
f(2)=3(2)^2-12(2)+9=3\cdot4-24+9=12-24+9=-3.
$$
So, $$ f(c)=-3 $$.

$$ c=2,\quad f(c)=-3. $$

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$$ \textbf{(b)} $$ Extreme values of $$ f $$ on $$[0,5]$$

1. Evaluate $$ f(x) $$ at the endpoints and at the critical point (if it lies in the interval):
   - $$ x=0 $$:
     $$
     f(0)=3(0)^2-12(0)+9=9.
     $$
   - $$ x=5 $$:
     $$
     f(5)=3(5)^2-12(5)+9=3\cdot25-60+9=75-60+9=24.
     $$
   - The previously found critical point $$ x=2 $$ is in $$[0,5]$$:
     $$
     f(2)=-3.
     $$

2. Compare:
   - Maximum value is $$\max\{9,24,-3\}=24$$ at $$ x=5 $$.
   - Minimum value is $$\min\{9,24,-3\}=-3$$ at $$ x=2 $$.

Maximum $$ =24, \quad\text{Minimum } =-3. $$

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$$ \textbf{(c)} $$ Extreme values of $$ f $$ on $$[-8,1]$$

1. Evaluate $$ f(x) $$ at the endpoints:
   - $$ x=-8 $$:
     $$
     f(-8)=3(-8)^2-12(-8)+9=3\cdot64+96+9=192+96+9=297.
     $$
   - $$ x=1 $$:
     $$
     f(1)=3(1)^2-12(1)+9=3-12+9=0.
     $$
   - The critical point $$ x=2 $$ does not lie in $$[-8,1]$$.

2. Compare:
   - Maximum value is $$\max\{297,0\}=297$$.
   - Minimum value is $$\min\{297,0\}=0$$.

Maximum $$ =297, \quad\text{Minimum } =0. $$
$$ \textbf{(a)} $$ - Critical point $$ c = 2 $$ - $$ f(c) = -3 $$ $$ \textbf{(b)} $$ - Maximum $$ = 24 $$ at $$ x = 5 $$ - Minimum $$ = -3 $$ at $$ x = 2 $$ $$ \textbf{(c)} $$ - Maximum $$ = 297 $$ at $$ x = -8 $$ - Minimum $$ = 0 $$ at $$ x = 1 $$

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