Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the \( x \)-values at which they occur.
\[
f(x)=7x+9
\]
(A) \( [0,9] \)
(B) \( [-4,5] \)
(A) Find the first derivative of \( f \).
\[
f'(x)=7
\]
(Simplify your answer.)
The absolute maximum value is \( \square \) at \( x=\square \).
(Use a comma to separate answers as needed.)
- Determine the critical points of \(f(x)\) within the interval [0,9] by solving \(f'(x) = 0\).
- There are no critical points because \(f'(x) = 7 \neq 0\).
- Evaluate \(f(x)\) at the endpoints of the interval [0,9] to find the function values.
- At \(x=0\), \(f(0) = 7(0) + 9 = 9\). At \(x=9\), \(f(9) = 7(9) + 9 = 72\).
- Compare the function values at the endpoints and any critical points within the interval [0,9] to determine the absolute maximum and minimum values.
- The absolute minimum value is 9 at \(x=0\), and the absolute maximum value is 72 at \(x=9\).
- Determine the critical points of \(f(x)\) within the interval [-4,5] by solving \(f'(x) = 0\).
- There are no critical points because \(f'(x) = 7 \neq 0\).
- Evaluate \(f(x)\) at the endpoints of the interval [-4,5] to find the function values.
- At \(x=-4\), \(f(-4) = 7(-4) + 9 = -28 + 9 = -19\). At \(x=5\), \(f(5) = 7(5) + 9 = 35 + 9 = 44\).
- Compare the function values at the endpoints and any critical points within the interval [-4,5] to determine the absolute maximum and minimum values.
- The absolute minimum value is -19 at \(x=-4\), and the absolute maximum value is 44 at \(x=5\).
