At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Sure! Let's explore the function [tex]\( y = 2x^3 - 3x^2 + 6x \)[/tex] step-by-step.
1. Identification of the Function:
The given function is a polynomial function of degree 3. It can be expressed as:
[tex]\[ y = 2x^3 - 3x^2 + 6x \][/tex]
2. Coefficients and Terms:
- The term [tex]\( 2x^3 \)[/tex] has a coefficient of 2 and represents the cubic term.
- The term [tex]\( -3x^2 \)[/tex] has a coefficient of -3 and represents the quadratic term.
- The term [tex]\( 6x \)[/tex] has a coefficient of 6 and represents the linear term.
3. First Derivative:
To analyze the behavior of the function, we may need to find the first derivative [tex]\( \frac{dy}{dx} \)[/tex]:
[tex]\[ \frac{dy}{dx} = \frac{d}{dx}(2x^3 - 3x^2 + 6x) = 6x^2 - 6x + 6 \][/tex]
4. Second Derivative:
The second derivative [tex]\( \frac{d^2y}{dx^2} \)[/tex] helps us understand the concavity of the function:
[tex]\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(6x^2 - 6x + 6) = 12x - 6 \][/tex]
5. Critical Points:
To find the critical points, we set the first derivative to zero:
[tex]\[ 6x^2 - 6x + 6 = 0 \][/tex]
Simplifying further, we get:
[tex]\[ x^2 - x + 1 = 0 \][/tex]
The discriminant of this quadratic equation is [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[ (-1)^2 - 4(1)(1) = 1 - 4 = -3 \][/tex]
Since the discriminant is negative, there are no real roots, and hence no real critical points.
6. Behavior and Graph Analysis:
- Since there are no real roots for the first derivative, the function does not have any turning points.
- The second derivative [tex]\( 12x - 6 \)[/tex] helps us understand the concavity:
- When [tex]\( x > \frac{1}{2} \)[/tex], [tex]\( 12x - 6 > 0 \)[/tex], indicating the function is concave up.
- When [tex]\( x < \frac{1}{2} \)[/tex], [tex]\( 12x - 6 < 0 \)[/tex], indicating the function is concave down.
7. Inflection Point:
The point where the concavity changes is known as the inflection point, found by setting the second derivative to zero:
[tex]\[ 12x - 6 = 0 \implies x = \frac{1}{2} \][/tex]
Substituting [tex]\( x = \frac{1}{2} \)[/tex] back into the original function to find [tex]\( y \)[/tex]:
[tex]\[ y = 2\left(\frac{1}{2}\right)^3 - 3\left(\frac{1}{2}\right)^2 + 6\left(\frac{1}{2}\right) = 2\left(\frac{1}{8}\right) - 3\left(\frac{1}{4}\right) + 3 = \frac{1}{4} - \frac{3}{4} + 3 = 2.5 \][/tex]
Thus, the inflection point is [tex]\( \left( \frac{1}{2}, 2.5 \right) \)[/tex].
8. Conclusion:
The function has an inflection point at [tex]\( \left( \frac{1}{2}, 2.5 \right) \)[/tex] and changes concavity at this point. The graph does not have any other turning points but exhibits only one concavity change.
So, the function [tex]\( y = 2x^3 - 3x^2 + 6x \)[/tex] has interesting features such as changing concavity at [tex]\( x = \frac{1}{2} \)[/tex] but no critical points. This is reflective of the function's polynomial nature.
1. Identification of the Function:
The given function is a polynomial function of degree 3. It can be expressed as:
[tex]\[ y = 2x^3 - 3x^2 + 6x \][/tex]
2. Coefficients and Terms:
- The term [tex]\( 2x^3 \)[/tex] has a coefficient of 2 and represents the cubic term.
- The term [tex]\( -3x^2 \)[/tex] has a coefficient of -3 and represents the quadratic term.
- The term [tex]\( 6x \)[/tex] has a coefficient of 6 and represents the linear term.
3. First Derivative:
To analyze the behavior of the function, we may need to find the first derivative [tex]\( \frac{dy}{dx} \)[/tex]:
[tex]\[ \frac{dy}{dx} = \frac{d}{dx}(2x^3 - 3x^2 + 6x) = 6x^2 - 6x + 6 \][/tex]
4. Second Derivative:
The second derivative [tex]\( \frac{d^2y}{dx^2} \)[/tex] helps us understand the concavity of the function:
[tex]\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(6x^2 - 6x + 6) = 12x - 6 \][/tex]
5. Critical Points:
To find the critical points, we set the first derivative to zero:
[tex]\[ 6x^2 - 6x + 6 = 0 \][/tex]
Simplifying further, we get:
[tex]\[ x^2 - x + 1 = 0 \][/tex]
The discriminant of this quadratic equation is [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[ (-1)^2 - 4(1)(1) = 1 - 4 = -3 \][/tex]
Since the discriminant is negative, there are no real roots, and hence no real critical points.
6. Behavior and Graph Analysis:
- Since there are no real roots for the first derivative, the function does not have any turning points.
- The second derivative [tex]\( 12x - 6 \)[/tex] helps us understand the concavity:
- When [tex]\( x > \frac{1}{2} \)[/tex], [tex]\( 12x - 6 > 0 \)[/tex], indicating the function is concave up.
- When [tex]\( x < \frac{1}{2} \)[/tex], [tex]\( 12x - 6 < 0 \)[/tex], indicating the function is concave down.
7. Inflection Point:
The point where the concavity changes is known as the inflection point, found by setting the second derivative to zero:
[tex]\[ 12x - 6 = 0 \implies x = \frac{1}{2} \][/tex]
Substituting [tex]\( x = \frac{1}{2} \)[/tex] back into the original function to find [tex]\( y \)[/tex]:
[tex]\[ y = 2\left(\frac{1}{2}\right)^3 - 3\left(\frac{1}{2}\right)^2 + 6\left(\frac{1}{2}\right) = 2\left(\frac{1}{8}\right) - 3\left(\frac{1}{4}\right) + 3 = \frac{1}{4} - \frac{3}{4} + 3 = 2.5 \][/tex]
Thus, the inflection point is [tex]\( \left( \frac{1}{2}, 2.5 \right) \)[/tex].
8. Conclusion:
The function has an inflection point at [tex]\( \left( \frac{1}{2}, 2.5 \right) \)[/tex] and changes concavity at this point. The graph does not have any other turning points but exhibits only one concavity change.
So, the function [tex]\( y = 2x^3 - 3x^2 + 6x \)[/tex] has interesting features such as changing concavity at [tex]\( x = \frac{1}{2} \)[/tex] but no critical points. This is reflective of the function's polynomial nature.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
