Find The Exact Location Of All The Relative And Absolute Extrema Of The Function. (Order Your Answers From Smallest To Largest $x$.)Given: $f(x) = X^2 - 4x + 1$ With Domain \[0, 3\].1. $f$ Has An Absolute Maximum At

Introduction

In calculus, extrema refer to the maximum or minimum values of a function. These values can be either absolute or relative. An absolute maximum is the highest value of a function within its domain, while a relative maximum is a local maximum that may not be the absolute maximum. In this article, we will find the exact location of all the relative and absolute extrema of the function $f(x) = x^2 - 4x + 1$ with a domain of $[0, 3]$.

Understanding the Function

The given function is a quadratic function in the form of $f(x) = ax^2 + bx + c$. In this case, $a = 1$, $b = -4$, and $c = 1$. The graph of a quadratic function is a parabola that opens upwards if $a > 0$ and downwards if $a < 0$. Since $a = 1$ in this case, the parabola opens upwards.

Finding the Vertex

The vertex of a parabola is the point where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum point of the function, depending on whether the parabola opens upwards or downwards. To find the vertex, we can use the formula:

$x = -\frac{b}{2a}$

Plugging in the values of $a$ and $b$, we get:

$x = -\frac{-4}{2(1)}$ $x = 2$

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging $x = 2$ into the function:

$f(2) = (2)^2 - 4(2) + 1$ $f(2) = 4 - 8 + 1$ $f(2) = -3$

So, the vertex of the parabola is at the point $(2, -3)$.

Finding the Absolute Maximum

Since the parabola opens upwards, the vertex is the minimum point of the function. However, we are interested in finding the absolute maximum of the function within the given domain $[0, 3]$. To do this, we need to check the values of the function at the endpoints of the domain and compare them with the value of the function at the vertex.

Checking the Endpoints

Let's evaluate the function at the endpoints of the domain:

$f(0) = (0)^2 - 4(0) + 1$ $f(0) = 1$

$f(3) = (3)^2 - 4(3) + 1$ $f(3) = 9 - 12 + 1$ $f(3) = -2$

Comparing the Values

Now that we have the values of the function at the endpoints and the vertex, we can compare them to find the absolute maximum:

$f(0) = 1$ $f(2) = -3$ $f(3) = -2$

The highest value of the function is $f(0) = 1$, which occurs at the endpoint $x = 0$. Therefore, the absolute maximum of the function is at the point $(0, 1)$.

Finding the Relative Maximum

Since the parabola opens upwards, there is no relative maximum within the given domain $[0, 3]$. However, if the domain were to extend beyond the vertex, there would be a relative maximum at the vertex.

Conclusion

In this article, we found the exact location of all the relative and absolute extrema of the function $f(x) = x^2 - 4x + 1$ with a domain of $[0, 3]$. We found that the absolute maximum of the function is at the point $(0, 1)$, and there is no relative maximum within the given domain.

Final Answer

The absolute maximum of the function is at the point $(0, 1)$.

Step-by-Step Solution

1. Find the vertex of the parabola using the formula $x = -\frac{b}{2a}$.
2. Evaluate the function at the vertex to find the y-coordinate.
3. Check the values of the function at the endpoints of the domain.
4. Compare the values of the function at the endpoints and the vertex to find the absolute maximum.
5. Since the parabola opens upwards, there is no relative maximum within the given domain.

Key Concepts

• Quadratic function
• Vertex of a parabola
• Absolute maximum
• Relative maximum
• Domain of a function

Mathematical Formulas

• $x = -\frac{b}{2a}$ (formula for finding the vertex)
• $f(x) = ax^2 + bx + c$ (general form of a quadratic function)

Graphical Representation

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about finding absolute and relative extrema of a quadratic function.

Q: What is the difference between an absolute maximum and a relative maximum?

A: An absolute maximum is the highest value of a function within its domain, while a relative maximum is a local maximum that may not be the absolute maximum.

Q: How do I find the absolute maximum of a quadratic function?

A: To find the absolute maximum of a quadratic function, you need to evaluate the function at the endpoints of the domain and compare the values with the value of the function at the vertex.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum point of the function, depending on whether the parabola opens upwards or downwards.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the values of the input variables that make the function undefined or undefined in the real number system.

Q: Can a quadratic function have more than one relative maximum?

A: Yes, a quadratic function can have more than one relative maximum if the parabola has multiple turning points.

Q: How do I find the relative maximum of a quadratic function?

A: To find the relative maximum of a quadratic function, you need to evaluate the function at the endpoints of the domain and compare the values with the value of the function at the vertex.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is significant because it represents the minimum or maximum point of the function, depending on whether the parabola opens upwards or downwards.

Q: Can a quadratic function have a relative maximum at the endpoint of the domain?

A: Yes, a quadratic function can have a relative maximum at the endpoint of the domain if the parabola opens upwards and the endpoint is a local maximum.

Q: How do I determine if a quadratic function has a relative maximum at the endpoint of the domain?

A: To determine if a quadratic function has a relative maximum at the endpoint of the domain, you need to evaluate the function at the endpoint and compare the value with the value of the function at the vertex.

Q: What is the relationship between the vertex and the relative maximum of a quadratic function?

A: The vertex and the relative maximum of a quadratic function are related in that the vertex represents the minimum or maximum point of the function, while the relative maximum represents a local maximum that may not be the absolute maximum.

Q: Can a quadratic function have a relative minimum at the endpoint of the domain?

A: Yes, a quadratic function can have a relative minimum at the endpoint of the domain if the parabola opens downwards and the endpoint is a local minimum.

Q: How do I determine if a quadratic function has a relative minimum at the endpoint of the domain?

A: To determine if a quadratic function has a relative minimum at the endpoint of the domain, you need to evaluate the function at the endpoint and compare the value with the value of the function at the vertex.

Conclusion

In this article, we have answered some of the most frequently asked questions about finding absolute and relative extrema of a quadratic function. We have discussed the difference between an absolute maximum and a relative maximum, how to find the absolute maximum of a quadratic function, and the significance of the vertex of a parabola. We have also answered questions about the domain of a function, the relationship between the vertex and the relative maximum of a quadratic function, and how to determine if a quadratic function has a relative maximum or minimum at the endpoint of the domain.

Final Answer

The absolute maximum of a quadratic function is at the point where the function is evaluated at the endpoints of the domain and compared with the value of the function at the vertex.

Step-by-Step Solution

1. Evaluate the function at the endpoints of the domain.
2. Compare the values of the function at the endpoints with the value of the function at the vertex.
3. Determine if the function has a relative maximum or minimum at the endpoint of the domain.
4. Use the formula $x = -\frac{b}{2a}$ to find the vertex of the parabola.
5. Evaluate the function at the vertex to find the y-coordinate.

Key Concepts

• Absolute maximum
• Relative maximum
• Vertex of a parabola
• Domain of a function
• Quadratic function

Mathematical Formulas

• $x = -\frac{b}{2a}$ (formula for finding the vertex)
• $f(x) = ax^2 + bx + c$ (general form of a quadratic function)

Graphical Representation

The graph of a quadratic function is a parabola that opens upwards or downwards. The vertex of the parabola represents the minimum or maximum point of the function, depending on whether the parabola opens upwards or downwards. The absolute maximum of a quadratic function is at the point where the function is evaluated at the endpoints of the domain and compared with the value of the function at the vertex.