Answer :
Let's analyze the function [tex]\( f(x) = -0.08 x (x^2 - 11x + 18) \)[/tex] to determine its critical points, and evaluate it at some points to understand the shape of its graph step-by-step.
### Step 1: Identify Critical Points
Critical points occur where the derivative [tex]\( f'(x) \)[/tex] is zero or undefined. For [tex]\( f(x) \)[/tex], let's find the points where the first derivative equals zero.
The critical points are:
[tex]\[ x = 0.938, \quad x = 6.395 \][/tex]
### Step 2: Evaluate [tex]\( f(x) \)[/tex] at Critical Points
To understand the behavior of the function at the critical points, we evaluate [tex]\( f(x) \)[/tex] at these points.
- At [tex]\( x = 0.938 \)[/tex]:
[tex]\[ f(0.938) = -0.642 \][/tex]
- At [tex]\( x = 6.395 \)[/tex]:
[tex]\[ f(6.395) = 5.857 \][/tex]
### Step 3: Evaluate [tex]\( f(x) \)[/tex] at Boundary Points
To further understand the function’s behavior, we evaluate the function at several boundary points within a typical range of [tex]\( x \)[/tex].
The boundary evaluations within the range [tex]\(-10 \leq x \leq 10\)[/tex] give us:
[tex]\[ f(-10) = 182.4, \quad f(-9) = 142.56, \quad f(-8) = 108.8, \quad f(-7) = 80.64, \quad f(-6) = 57.6, \quad f(-5) = 39.2, \quad f(-4) = 24.96, \quad f(-3) = 14.4, \quad f(-2) = 7.04, \quad f(-1) = 2.4, \quad f(0) = 0, \][/tex]
[tex]\[ f(1) = -0.64, \quad f(2) = 0, \quad f(3) = 1.44, \quad f(4) = 3.2, \quad f(5) = 4.8, \quad f(6) = 5.76, \quad f(7) = 5.6, \quad f(8) = 3.84, \quad f(9) = 0, \quad f(10) = -6.4 \][/tex]
### Step 4: Analyze and Graph the Function
From the evaluations and critical points:
- The function starts decreasing from high positive values in the far negative [tex]\( x \)[/tex] range.
- It crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex], and slightly below the axis at [tex]\( x \approx 1 \)[/tex].
- There's a local minimum around [tex]\( x = 0.938 \)[/tex] where [tex]\( f(0.938) = -0.642 \)[/tex].
- The function continues increasing to a local maximum at [tex]\( x = 6.395 \)[/tex] where [tex]\( f(6.395) = 5.857 \)[/tex].
- After that, the function gradually decreases, reaching [tex]\( y = 0 \)[/tex] again at [tex]\( x \approx 9 \)[/tex], and continues decreasing into negative values towards [tex]\( x = 10 \)[/tex].
### Conclusion
The shape of the function exhibits a local minimum around [tex]\( x = 1 \)[/tex], a significant local maximum around [tex]\( x = 6.395 \)[/tex], then declines as [tex]\( x \)[/tex] approaches 10.
To choose the correct graph, look for:
1. High positive values for negative [tex]\( x \)[/tex].
2. Crossing the x-axis at [tex]\( x = 0 \)[/tex].
3. Local minimum near [tex]\( x = 0.938 \)[/tex].
4. Local maximum near [tex]\( x = 6.395 \)[/tex].
5. Function values becoming negative again towards [tex]\( x = 10 \)[/tex].
Given this behavior, the graph will have this specific pattern of turning points and bounds. Ensure the graph you choose matches these characteristics clearly.
### Step 1: Identify Critical Points
Critical points occur where the derivative [tex]\( f'(x) \)[/tex] is zero or undefined. For [tex]\( f(x) \)[/tex], let's find the points where the first derivative equals zero.
The critical points are:
[tex]\[ x = 0.938, \quad x = 6.395 \][/tex]
### Step 2: Evaluate [tex]\( f(x) \)[/tex] at Critical Points
To understand the behavior of the function at the critical points, we evaluate [tex]\( f(x) \)[/tex] at these points.
- At [tex]\( x = 0.938 \)[/tex]:
[tex]\[ f(0.938) = -0.642 \][/tex]
- At [tex]\( x = 6.395 \)[/tex]:
[tex]\[ f(6.395) = 5.857 \][/tex]
### Step 3: Evaluate [tex]\( f(x) \)[/tex] at Boundary Points
To further understand the function’s behavior, we evaluate the function at several boundary points within a typical range of [tex]\( x \)[/tex].
The boundary evaluations within the range [tex]\(-10 \leq x \leq 10\)[/tex] give us:
[tex]\[ f(-10) = 182.4, \quad f(-9) = 142.56, \quad f(-8) = 108.8, \quad f(-7) = 80.64, \quad f(-6) = 57.6, \quad f(-5) = 39.2, \quad f(-4) = 24.96, \quad f(-3) = 14.4, \quad f(-2) = 7.04, \quad f(-1) = 2.4, \quad f(0) = 0, \][/tex]
[tex]\[ f(1) = -0.64, \quad f(2) = 0, \quad f(3) = 1.44, \quad f(4) = 3.2, \quad f(5) = 4.8, \quad f(6) = 5.76, \quad f(7) = 5.6, \quad f(8) = 3.84, \quad f(9) = 0, \quad f(10) = -6.4 \][/tex]
### Step 4: Analyze and Graph the Function
From the evaluations and critical points:
- The function starts decreasing from high positive values in the far negative [tex]\( x \)[/tex] range.
- It crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex], and slightly below the axis at [tex]\( x \approx 1 \)[/tex].
- There's a local minimum around [tex]\( x = 0.938 \)[/tex] where [tex]\( f(0.938) = -0.642 \)[/tex].
- The function continues increasing to a local maximum at [tex]\( x = 6.395 \)[/tex] where [tex]\( f(6.395) = 5.857 \)[/tex].
- After that, the function gradually decreases, reaching [tex]\( y = 0 \)[/tex] again at [tex]\( x \approx 9 \)[/tex], and continues decreasing into negative values towards [tex]\( x = 10 \)[/tex].
### Conclusion
The shape of the function exhibits a local minimum around [tex]\( x = 1 \)[/tex], a significant local maximum around [tex]\( x = 6.395 \)[/tex], then declines as [tex]\( x \)[/tex] approaches 10.
To choose the correct graph, look for:
1. High positive values for negative [tex]\( x \)[/tex].
2. Crossing the x-axis at [tex]\( x = 0 \)[/tex].
3. Local minimum near [tex]\( x = 0.938 \)[/tex].
4. Local maximum near [tex]\( x = 6.395 \)[/tex].
5. Function values becoming negative again towards [tex]\( x = 10 \)[/tex].
Given this behavior, the graph will have this specific pattern of turning points and bounds. Ensure the graph you choose matches these characteristics clearly.
