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Sagot :
To analyze the given function \( f(x) = x^3 - 2x^2 - 19x + 20 \) and determine which statements correctly describe the graph, we'll go through each option step-by-step.
1. As the \( x \)-values increase, the \( y \)-values always increase.
To check if the function is always increasing, we need to take the first derivative \( f'(x) \) and analyze its sign.
[tex]\[ f'(x) = \frac{d}{dx} \left( x^3 - 2x^2 - 19x + 20 \right) = 3x^2 - 4x - 19 \][/tex]
Next, we find the critical points by setting the first derivative to zero:
[tex]\[ 3x^2 - 4x - 19 = 0 \][/tex]
Solving this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
[tex]\[ x = \frac{4 \pm \sqrt{16 + 228}}{6} = \frac{4 \pm \sqrt{244}}{6} = \frac{4 \pm 2\sqrt{61}}{6} = \frac{2 \pm \sqrt{61}}{3} \][/tex]
Since the quadratic has real roots, it means that \( f'(x) \) changes sign, indicating the function has local minima and maxima and is not always increasing. Thus, the first option is incorrect.
2. As \( x \) approaches negative infinity, \( y \) approaches negative infinity.
For a cubic polynomial \( ax^3 + bx^2 + cx + d \), the end behavior is dominated by the term \( ax^3 \). Since \( a = 1 \), as \( x \to -\infty \), \( f(x) \to -\infty \). This statement is correct.
3. The domain of the function is all real numbers.
Polynomials are defined for all real numbers, so the domain is \( (-\infty, \infty) \). This statement is correct.
4. The range of the function is \( y \geq 20 \).
To determine the range, consider that a cubic function typically has a range of all real numbers because it can stretch infinitely in both the positive and negative directions. Therefore, this statement is incorrect.
5. The graph has a positive \( y \)-intercept.
The \( y \)-intercept is found by evaluating the function at \( x = 0 \):
[tex]\[ f(0) = 0^3 - 2(0)^2 - 19(0) + 20 = 20 \][/tex]
Since 20 is positive, this statement is correct.
Summarizing the statements:
- As \( x \) approaches negative infinity, \( y \) approaches negative infinity: True
- The domain of the function is all real numbers: True
- The graph has a positive \( y \)-intercept: True
The three correct statements are:
2. As \( x \) approaches negative infinity, \( y \) approaches negative infinity.
3. The domain of the function is all real numbers.
5. The graph has a positive [tex]\( y \)[/tex]-intercept.
1. As the \( x \)-values increase, the \( y \)-values always increase.
To check if the function is always increasing, we need to take the first derivative \( f'(x) \) and analyze its sign.
[tex]\[ f'(x) = \frac{d}{dx} \left( x^3 - 2x^2 - 19x + 20 \right) = 3x^2 - 4x - 19 \][/tex]
Next, we find the critical points by setting the first derivative to zero:
[tex]\[ 3x^2 - 4x - 19 = 0 \][/tex]
Solving this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
[tex]\[ x = \frac{4 \pm \sqrt{16 + 228}}{6} = \frac{4 \pm \sqrt{244}}{6} = \frac{4 \pm 2\sqrt{61}}{6} = \frac{2 \pm \sqrt{61}}{3} \][/tex]
Since the quadratic has real roots, it means that \( f'(x) \) changes sign, indicating the function has local minima and maxima and is not always increasing. Thus, the first option is incorrect.
2. As \( x \) approaches negative infinity, \( y \) approaches negative infinity.
For a cubic polynomial \( ax^3 + bx^2 + cx + d \), the end behavior is dominated by the term \( ax^3 \). Since \( a = 1 \), as \( x \to -\infty \), \( f(x) \to -\infty \). This statement is correct.
3. The domain of the function is all real numbers.
Polynomials are defined for all real numbers, so the domain is \( (-\infty, \infty) \). This statement is correct.
4. The range of the function is \( y \geq 20 \).
To determine the range, consider that a cubic function typically has a range of all real numbers because it can stretch infinitely in both the positive and negative directions. Therefore, this statement is incorrect.
5. The graph has a positive \( y \)-intercept.
The \( y \)-intercept is found by evaluating the function at \( x = 0 \):
[tex]\[ f(0) = 0^3 - 2(0)^2 - 19(0) + 20 = 20 \][/tex]
Since 20 is positive, this statement is correct.
Summarizing the statements:
- As \( x \) approaches negative infinity, \( y \) approaches negative infinity: True
- The domain of the function is all real numbers: True
- The graph has a positive \( y \)-intercept: True
The three correct statements are:
2. As \( x \) approaches negative infinity, \( y \) approaches negative infinity.
3. The domain of the function is all real numbers.
5. The graph has a positive [tex]\( y \)[/tex]-intercept.
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