find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated…

Explanation:

Step1: Find the derivative

The derivative of $f(x)=24x - x^{2}$ is $f^\prime(x)=24 - 2x$.

Step2: Set derivative equal to 0

Set $24 - 2x = 0$. Solving for $x$ gives $2x=24$, so $x = 12$.

Step3: Determine if it's a max or min

The second - derivative $f^{\prime\prime}(x)=-2<0$. So $x = 12$ is a local maximum.

Step4: Find the maximum value

Substitute $x = 12$ into $f(x)$: $f(12)=24\times12-12^{2}=288 - 144=144$. Since the function is a parabola opening downwards ($a=-1$ in $y = ax^{2}+bx + c$), as $x\to\pm\infty$, $y\to-\infty$. So there is no absolute minimum.

Answer:

C. Absolute maximum is 144; no absolute minimum