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)=3x + 8
(a)0,5
(b)-3,3
(a) find the first derivative of f.
f(x)=\square
(simplify your answer.)
the absolute maximum value is \square at x = \square.
(use a comma to separate answers as needed.)
the absolute minimum value is \square at x = \square.
(use a comma to separate answers as needed.)
(b) the absolute maximum value is \square at x = \square.
(use a comma to separate answers as needed.)
the absolute minimum value is \square at x = \square.
(use a comma to separate answers as needed.)

Explanation:

Step1: Find the first derivative

Using the power rule \((x^n)^\prime=nx^{n - 1}\), for \(y = 3x+8\), \(f^\prime(x)=\frac{d}{dx}(3x)+\frac{d}{dx}(8)\). Since \(\frac{d}{dx}(ax)=a\) (\(a = 3\) here) and \(\frac{d}{dx}(c)=0\) (\(c = 8\) here), \(f^\prime(x)=3\).

Step2: Analyze the function on the interval \([0,5]\)

Since \(f^\prime(x)=3>0\), the function \(y = f(x)=3x + 8\) is increasing on the interval \([0,5]\).

• For the absolute maximum:

Evaluate \(f(x)\) at the endpoints. \(f(0)=3\times0 + 8=8\) and \(f(5)=3\times5+8=15 + 8=23\). The absolute maximum value is \(23\) at \(x = 5\).

• For the absolute minimum:

Evaluate \(f(x)\) at the endpoints. \(f(0)=8\) and \(f(5)=23\). The absolute minimum value is \(8\) at \(x = 0\).

Step3: Analyze the function on the interval \([-3,3]\)

Since \(f^\prime(x)=3>0\), the function \(y = f(x)=3x + 8\) is increasing on the interval \([-3,3]\).

• For the absolute maximum:

Evaluate \(f(x)\) at the endpoints. \(f(-3)=3\times(-3)+8=-9 + 8=-1\) and \(f(3)=3\times3+8=9 + 8=17\). The absolute maximum value is \(17\) at \(x = 3\).

• For the absolute minimum:

Evaluate \(f(x)\) at the endpoints. \(f(-3)=-1\) and \(f(3)=17\). The absolute minimum value is \(-1\) at \(x=-3\).

Answer:

(A) \(f^\prime(x)=3\)
(A) Absolute maximum: \(23\) at \(x = 5\); Absolute minimum: \(8\) at \(x = 0\)
(B) Absolute maximum: \(17\) at \(x = 3\); Absolute minimum: \(-1\) at \(x=-3\)