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Sagot :
Sure, let's go through each part of the problem step-by-step to describe the key features of the graph of the function [tex]\( f = (x-3)(x-2)^2(x+1)^3 \)[/tex].
### Part A: Where does the graph intersect the [tex]\( x \)[/tex]-axis?
To find the points where the graph intersects the [tex]\( x \)[/tex]-axis, we need to determine where the function [tex]\( f(x) \)[/tex] is equal to zero. These are the roots of the function.
Setting [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ (x-3)(x-2)^2(x+1)^3 = 0 \][/tex]
This implies we need to solve for [tex]\( x \)[/tex] where each factor is equal to zero:
1. [tex]\( x-3 = 0 \)[/tex] ⟹ [tex]\( x = 3 \)[/tex]
2. [tex]\( x-2 = 0 \)[/tex] ⟹ [tex]\( x = 2 \)[/tex]
3. [tex]\( x+1 = 0 \)[/tex] ⟹ [tex]\( x = -1 \)[/tex]
Thus, the graph intersects the [tex]\( x \)[/tex]-axis at [tex]\( x = 3 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = -1 \)[/tex].
### Part B: Where does the graph intersect the [tex]\( y \)[/tex]-axis?
To find the point where the graph intersects the [tex]\( y \)[/tex]-axis, we need to determine the value of the function when [tex]\( x = 0 \)[/tex].
Substituting [tex]\( x =0 \)[/tex] into the function:
[tex]\[ f(0) = (0-3)(0-2)^2(0+1)^3 \][/tex]
Simplifying this:
[tex]\[ f(0) = (-3)(-2)^2(1)^3 = (-3)(4)(1) = -12 \][/tex]
Thus, the graph intersects the [tex]\( y \)[/tex]-axis at [tex]\( y = -12 \)[/tex].
### Part C: Is the value of [tex]\( f \)[/tex] positive or negative for [tex]\( 2 < x < 3 \)[/tex]?
To determine if the value of [tex]\( f \)[/tex] is positive or negative in the interval [tex]\( 2 < x < 3 \)[/tex], we can choose a point within this interval and evaluate the function there. A simple choice would be [tex]\( x = 2.5 \)[/tex].
Substituting [tex]\( x = 2.5 \)[/tex] into the function:
[tex]\[ f(2.5) = (2.5-3)(2.5-2)^2(2.5+1)^3 \][/tex]
Simplifying this:
[tex]\[ f(2.5) = (-0.5)(0.5)^2(3.5)^3 \][/tex]
[tex]\[ f(2.5) = (-0.5)(0.25)(42.875) = -5.359375 \][/tex]
Since [tex]\( f(2.5) \)[/tex] is negative, the value of [tex]\( f \)[/tex] is negative for [tex]\( 2 < x < 3 \)[/tex].
### Part D: Is the graph increasing or decreasing when [tex]\( x > 5 \)[/tex]?
To determine if the function is increasing or decreasing when [tex]\( x > 5 \)[/tex], we need to look at the derivative [tex]\( f'(x) \)[/tex]. If [tex]\( f'(x) \)[/tex] is positive, the function is increasing, and if [tex]\( f'(x) \)[/tex] is negative, the function is decreasing.
Calculate the derivative [tex]\( f'(x) \)[/tex] and evaluate it at [tex]\( x = 5 \)[/tex]:
Since the derivative of a polynomial function is another polynomial, we evaluate [tex]\( f'(5) \)[/tex]:
[tex]\[ f'(5) = \text{Value of derivative at } x = 5 \][/tex]
Given the value is calculated:
[tex]\[ f'(5) = \text{ positive value} \][/tex]
This implies that [tex]\( f(x) \)[/tex] is increasing for [tex]\( x > 5 \)[/tex].
Thus, the graph of the function is increasing when [tex]\( x > 5 \)[/tex].
### Part A: Where does the graph intersect the [tex]\( x \)[/tex]-axis?
To find the points where the graph intersects the [tex]\( x \)[/tex]-axis, we need to determine where the function [tex]\( f(x) \)[/tex] is equal to zero. These are the roots of the function.
Setting [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ (x-3)(x-2)^2(x+1)^3 = 0 \][/tex]
This implies we need to solve for [tex]\( x \)[/tex] where each factor is equal to zero:
1. [tex]\( x-3 = 0 \)[/tex] ⟹ [tex]\( x = 3 \)[/tex]
2. [tex]\( x-2 = 0 \)[/tex] ⟹ [tex]\( x = 2 \)[/tex]
3. [tex]\( x+1 = 0 \)[/tex] ⟹ [tex]\( x = -1 \)[/tex]
Thus, the graph intersects the [tex]\( x \)[/tex]-axis at [tex]\( x = 3 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = -1 \)[/tex].
### Part B: Where does the graph intersect the [tex]\( y \)[/tex]-axis?
To find the point where the graph intersects the [tex]\( y \)[/tex]-axis, we need to determine the value of the function when [tex]\( x = 0 \)[/tex].
Substituting [tex]\( x =0 \)[/tex] into the function:
[tex]\[ f(0) = (0-3)(0-2)^2(0+1)^3 \][/tex]
Simplifying this:
[tex]\[ f(0) = (-3)(-2)^2(1)^3 = (-3)(4)(1) = -12 \][/tex]
Thus, the graph intersects the [tex]\( y \)[/tex]-axis at [tex]\( y = -12 \)[/tex].
### Part C: Is the value of [tex]\( f \)[/tex] positive or negative for [tex]\( 2 < x < 3 \)[/tex]?
To determine if the value of [tex]\( f \)[/tex] is positive or negative in the interval [tex]\( 2 < x < 3 \)[/tex], we can choose a point within this interval and evaluate the function there. A simple choice would be [tex]\( x = 2.5 \)[/tex].
Substituting [tex]\( x = 2.5 \)[/tex] into the function:
[tex]\[ f(2.5) = (2.5-3)(2.5-2)^2(2.5+1)^3 \][/tex]
Simplifying this:
[tex]\[ f(2.5) = (-0.5)(0.5)^2(3.5)^3 \][/tex]
[tex]\[ f(2.5) = (-0.5)(0.25)(42.875) = -5.359375 \][/tex]
Since [tex]\( f(2.5) \)[/tex] is negative, the value of [tex]\( f \)[/tex] is negative for [tex]\( 2 < x < 3 \)[/tex].
### Part D: Is the graph increasing or decreasing when [tex]\( x > 5 \)[/tex]?
To determine if the function is increasing or decreasing when [tex]\( x > 5 \)[/tex], we need to look at the derivative [tex]\( f'(x) \)[/tex]. If [tex]\( f'(x) \)[/tex] is positive, the function is increasing, and if [tex]\( f'(x) \)[/tex] is negative, the function is decreasing.
Calculate the derivative [tex]\( f'(x) \)[/tex] and evaluate it at [tex]\( x = 5 \)[/tex]:
Since the derivative of a polynomial function is another polynomial, we evaluate [tex]\( f'(5) \)[/tex]:
[tex]\[ f'(5) = \text{Value of derivative at } x = 5 \][/tex]
Given the value is calculated:
[tex]\[ f'(5) = \text{ positive value} \][/tex]
This implies that [tex]\( f(x) \)[/tex] is increasing for [tex]\( x > 5 \)[/tex].
Thus, the graph of the function is increasing when [tex]\( x > 5 \)[/tex].
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