Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the $$ x $$-values at which they occur. $$ f(x)=8 x-4 $$ $$ \begin{array}{ll} ext { (A) }[0,5] & ext { (B) }[-6,5]\end{array} $$ (A) Find the first derivative of $$ f $$. $$ f^{\prime}(x)=8 $$ (Simplify your answer.) The absolute maximum value is $$ \square $$ at $$ x=\square $$.

Question

Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the $$ x $$-values at which they occur. $$ f(x)=8 x-4 $$ $$ \begin{array}{ll}	ext { (A) }[0,5] & 	ext { (B) }[-6,5]\end{array} $$ (A) Find the first derivative of $$ f $$. $$ f^{\prime}(x)=8 $$ (Simplify your answer.) The absolute maximum value is $$ \square $$ at $$ x=\square $$.

UpStudy AI · Accepted Answer

**Step 1. Compute the first derivative:**

The function is given by  
$$
f(x)=8x-4.
$$  
The first derivative of $$ f(x) $$ is  
$$
f'(x)=8.
$$

**Step 2. Identify critical points:**

Since $$ f'(x)=8 $$ is constant and never zero, there are no critical points in the open interval. Therefore, we only need to examine the endpoints of the interval $$[0,5]$$.

**Step 3. Evaluate $$ f(x) $$ at the endpoints:**

- At $$ x=0 $$:  
  $$
  f(0)=8(0)-4=-4.
  $$

- At $$ x=5 $$:  
  $$
  f(5)=8(5)-4=40-4=36.
  $$

**Step 4. Determine the absolute maximum and minimum:**

- The absolute maximum value is $$\boxed{36}$$ at $$ x=\boxed{5} $$.
- The absolute minimum value is $$\boxed{-4}$$ at $$ x=\boxed{0} $$.

Thus, for the interval $$[0,5]$$, the absolute maximum value is $$36$$ at $$x=5$$.
The absolute maximum value is 36 at $$ x=5 $$, and the absolute minimum value is -4 at $$ x=0 $$.

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