find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x…

find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x - values at which they occur.\nf(x)=3x + 8\n(a)0,5\n(b)-3,3\n(a) find the first derivative of f.\nf(x)=\\square\n(simplify your answer.)\nthe absolute maximum value is \\square at x = \\square.\n(use a comma to separate answers as needed.)\nthe absolute minimum value is \\square at x = \\square.\n(use a comma to separate answers as needed.)\n(b) the absolute maximum value is \\square at x = \\square.\n(use a comma to separate answers as needed.)\nthe absolute minimum value is \\square at x = \\square.\n(use a comma to separate answers as needed.)

find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x - values at which they occur.\nf(x)=3x + 8\n(a)0,5\n(b)-3,3\n(a) find the first derivative of f.\nf(x)=\\square\n(simplify your answer.)\nthe absolute maximum value is \\square at x = \\square.\n(use a comma to separate answers as needed.)\nthe absolute minimum value is \\square at x = \\square.\n(use a comma to separate answers as needed.)\n(b) the absolute maximum value is \\square at x = \\square.\n(use a comma to separate answers as needed.)\nthe absolute minimum value is \\square at x = \\square.\n(use a comma to separate answers as needed.)

Answer

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)