Find The Critical Numbers Of The Function F ( X ) = 3 X 5 − 15 X 4 − 5 X 3 + 4 F(x) = 3x^5 - 15x^4 - 5x^3 + 4 F ( X ) = 3 X 5 − 15 X 4 − 5 X 3 + 4 And Classify Them.At X = □ X = \square X = □ , The Function Has A □ \square □ . Select An Answer.At X = □ X = \square X = □ , The Function Has A □ \square □ .

Introduction

In calculus, critical numbers are the values of the variable that make the derivative of a function equal to zero or undefined. These points are crucial in understanding the behavior of a function, as they can indicate the presence of local maxima, minima, or points of inflection. In this article, we will delve into the concept of critical numbers and apply it to the polynomial function $f(x) = 3x^5 - 15x^4 - 5x^3 + 4$. We will find the critical numbers of the function, classify them, and analyze their significance.

The Derivative of the Function

To find the critical numbers of the function $f(x) = 3x^5 - 15x^4 - 5x^3 + 4$, we need to calculate its derivative. Using the power rule of differentiation, we get:

$$f'(x) = 15x^4 - 60x^3 - 15x^2$$

Finding Critical Numbers

Critical numbers occur when the derivative of a function is equal to zero or undefined. In this case, we need to find the values of $x$ that make $f'(x)$ equal to zero. We can do this by setting $f'(x)$ equal to zero and solving for $x$:

$$15x^4 - 60x^3 - 15x^2 = 0$$

We can factor out $15x^2$ from the equation:

$$15x^2(x^2 - 4x - 1) = 0$$

This gives us two possible solutions:

$$15x^2 = 0 \quad \text{or} \quad x^2 - 4x - 1 = 0$$

Solving for $x$ in the first equation, we get:

$$x^2 = 0 \quad \Rightarrow \quad x = 0$$

For the second equation, we can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -4$, and $c = -1$. Plugging these values into the formula, we get:

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

Simplifying the expression, we get:

$$x = \frac{4 \pm \sqrt{16 + 4}}{2} \quad \Rightarrow \quad x = \frac{4 \pm \sqrt{20}}{2}$$

$$x = \frac{4 \pm 2\sqrt{5}}{2} \quad \Rightarrow \quad x = 2 \pm \sqrt{5}$$

Classifying Critical Numbers

Now that we have found the critical numbers of the function, we need to classify them. To do this, we need to examine the behavior of the function around each critical number. We can do this by analyzing the sign of the derivative of the function in the intervals surrounding each critical number.

For $x = 0$, we can examine the sign of $f'(x)$ in the intervals $(-\infty, 0)$ and $(0, \infty)$. We can do this by plugging in values of $x$ into the derivative:

$$f'(-1) = 15(-1)^4 - 60(-1)^3 - 15(-1)^2 = 15 + 60 - 15 = 60$$

$$f'(1) = 15(1)^4 - 60(1)^3 - 15(1)^2 = 15 - 60 - 15 = -60$$

Since $f'(-1) > 0$ and $f'(1) < 0$, we can conclude that the function is increasing on the interval $(-\infty, 0)$ and decreasing on the interval $(0, \infty)$. This means that $x = 0$ is a local maximum.

For $x = 2 + \sqrt{5}$, we can examine the sign of $f'(x)$ in the intervals $(1, 2 + \sqrt{5})$ and $(2 + \sqrt{5}, \infty)$. We can do this by plugging in values of $x$ into the derivative:

$$f'(2) = 15(2)^4 - 60(2)^3 - 15(2)^2 = 240 - 480 - 60 = -300$$

$$f'(3) = 15(3)^4 - 60(3)^3 - 15(3)^2 = 1215 - 1620 - 135 = -540$$

Since $f'(2) < 0$ and $f'(3) < 0$, we can conclude that the function is decreasing on the interval $(1, 2 + \sqrt{5})$ and $(2 + \sqrt{5}, \infty)$. This means that $x = 2 + \sqrt{5}$ is a local minimum.

For $x = 2 - \sqrt{5}$, we can examine the sign of $f'(x)$ in the intervals $(2 - \sqrt{5}, 1)$ and $(1, 2 - \sqrt{5})$. We can do this by plugging in values of $x$ into the derivative:

$$f'(1) = 15(1)^4 - 60(1)^3 - 15(1)^2 = 15 - 60 - 15 = -60$$

$$f'(0) = 15(0)^4 - 60(0)^3 - 15(0)^2 = 0$$

Since $f'(1) < 0$ and $f'(0) = 0$, we can conclude that the function is decreasing on the interval $(2 - \sqrt{5}, 1)$ and increasing on the interval $(1, 2 - \sqrt{5})$. This means that $x = 2 - \sqrt{5}$ is a local minimum.

Conclusion

In this article, we have found the critical numbers of the function $f(x) = 3x^5 - 15x^4 - 5x^3 + 4$ and classified them. We have shown that the function has three critical numbers: $x = 0$, $x = 2 + \sqrt{5}$, and $x = 2 - \sqrt{5}$. We have also classified these critical numbers as local maxima and minima. The critical numbers of a function are crucial in understanding the behavior of the function, and they can be used to identify the presence of local maxima, minima, or points of inflection.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] A First Course in Calculus, 7th edition, Serge Lang

Glossary

• Critical number: A value of the variable that makes the derivative of a function equal to zero or undefined.
• Local maximum: A point on the graph of a function where the function changes from increasing to decreasing.
• Local minimum: A point on the graph of a function where the function changes from decreasing to increasing.
• Point of inflection: A point on the graph of a function where the function changes from concave up to concave down or vice versa.
  

Introduction

In our previous article, we explored the concept of critical numbers and applied it to the polynomial function $f(x) = 3x^5 - 15x^4 - 5x^3 + 4$. We found the critical numbers of the function, classified them, and analyzed their significance. In this article, we will address some of the most frequently asked questions related to critical numbers and provide additional insights into the topic.

Q&A

Q: What is the difference between a critical number and a local maximum/minima?

A: A critical number is a value of the variable that makes the derivative of a function equal to zero or undefined. A local maximum or minimum, on the other hand, is a point on the graph of a function where the function changes from increasing to decreasing or vice versa. While all local maxima and minima are critical numbers, not all critical numbers are local maxima or minima.

Q: How do I determine if a critical number is a local maximum or minimum?

A: To determine if a critical number is a local maximum or minimum, you need to examine the behavior of the function around that point. You can do this by analyzing the sign of the derivative of the function in the intervals surrounding the critical number. If the derivative changes from positive to negative, the critical number is a local maximum. If the derivative changes from negative to positive, the critical number is a local minimum.

Q: Can a critical number be a point of inflection?

A: Yes, a critical number can be a point of inflection. A point of inflection is a point on the graph of a function where the function changes from concave up to concave down or vice versa. If the derivative of a function changes sign at a critical number, that critical number can be a point of inflection.

Q: How do I find the critical numbers of a function?

A: To find the critical numbers of a function, you need to calculate the derivative of the function and set it equal to zero. You can then solve for the values of the variable that make the derivative equal to zero. These values are the critical numbers of the function.

Q: Can a function have multiple critical numbers?

A: Yes, a function can have multiple critical numbers. In fact, most functions have multiple critical numbers. The number of critical numbers a function has depends on the degree of the function and the complexity of its derivative.

Q: How do I classify the critical numbers of a function?

A: To classify the critical numbers of a function, you need to examine the behavior of the function around each critical number. You can do this by analyzing the sign of the derivative of the function in the intervals surrounding each critical number. If the derivative changes from positive to negative, the critical number is a local maximum. If the derivative changes from negative to positive, the critical number is a local minimum.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to critical numbers and provided additional insights into the topic. We have shown that critical numbers are crucial in understanding the behavior of a function and can be used to identify the presence of local maxima, minima, or points of inflection. By following the steps outlined in this article, you can find the critical numbers of a function and classify them.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] A First Course in Calculus, 7th edition, Serge Lang

Glossary

• Critical number: A value of the variable that makes the derivative of a function equal to zero or undefined.
• Local maximum: A point on the graph of a function where the function changes from increasing to decreasing.
• Local minimum: A point on the graph of a function where the function changes from decreasing to increasing.
• Point of inflection: A point on the graph of a function where the function changes from concave up to concave down or vice versa.