select all correct answers.\nthe graph of the first derivative f(x) of a function f is shown below. at what…

select all correct answers.\nthe graph of the first derivative f(x) of a function f is shown below. at what values of x does f have a local minimum?\n(description: the graph is titled y = f prime (x). a curve is graphed on the x y coordinate plane. the curve enters the top left of the viewing window, goes down and to the right, crosses the x - axis at (3, 0), reaches a low point near (4, negative 25), goes up and to the right, crosses the x - axis at (5, 0), reaches a high point near (6.5, 25), goes down and to the right, crosses the x - axis at (8, 0), reaches a low point near (9.5, negative 25), goes up and to the right, crosses the x - axis at (10, 0), and exits the top right of viewing window.)\n10\n9\n5\n8\n4\n3

select all correct answers.\nthe graph of the first derivative f(x) of a function f is shown below. at what values of x does f have a local minimum?\n(description: the graph is titled y = f prime (x). a curve is graphed on the x y coordinate plane. the curve enters the top left of the viewing window, goes down and to the right, crosses the x - axis at (3, 0), reaches a low point near (4, negative 25), goes up and to the right, crosses the x - axis at (5, 0), reaches a high point near (6.5, 25), goes down and to the right, crosses the x - axis at (8, 0), reaches a low point near (9.5, negative 25), goes up and to the right, crosses the x - axis at (10, 0), and exits the top right of viewing window.)\n10\n9\n5\n8\n4\n3

Answer

Explanation:

Step1: Recall the first - derivative test

The first - derivative test states that if (f^{\prime}(x)) changes sign from negative to positive at a point (x = a), then (f(x)) has a local minimum at (x=a).

Step2: Analyze the sign of (f^{\prime}(x)) around each critical point

  • For (x = 3):
    • To the left of (x = 3), (f^{\prime}(x)>0) (since the curve is above the (x) - axis for (x<3) in the relevant domain).
    • To the right of (x = 3), (f^{\prime}(x)<0) (since the curve is below the (x) - axis for (3 < x<5)). So (f(x)) has a local maximum at (x = 3).
  • For (x=5):
    • To the left of (x = 5), (f^{\prime}(x)<0) (for (3 < x<5)).
    • To the right of (x = 5), (f^{\prime}(x)>0) (for (5 < x<8)). So (f(x)) has a local minimum at (x = 5).
  • For (x = 8):
    • To the left of (x = 8), (f^{\prime}(x)>0) (for (5 < x<8)).
    • To the right of (x = 8), (f^{\prime}(x)<0) (for (8 < x<10)). So (f(x)) has a local maximum at (x = 8).
  • For (x = 10):
    • To the left of (x = 10), (f^{\prime}(x)<0) (for (8 < x<10)).
    • To the right of (x = 10), (f^{\prime}(x)>0) (as (x>10) and the function is increasing). So (f(x)) has a local minimum at (x = 10).

Answer:

5, 10