Finding the Local Minimum and Maximum

Oct 13, 2025 by windailys.com 15 views

Introduction

In mathematics, a cubic polynomial is a polynomial of degree three, which means the highest power of the variable is three. The general form of a cubic polynomial is $f(x) = ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants. In this article, we will discuss the function $f(x)=-2x^3+42x^2-270x+11$ and its local minimum and maximum.

Local Minimum and Maximum

A local minimum is a point on the graph of a function where the function value is smaller than or equal to the values at nearby points. Similarly, a local maximum is a point on the graph of a function where the function value is larger than or equal to the values at nearby points. In the case of the function $f(x)=-2x^3+42x^2-270x+11$ , we are given that it has one local minimum and one local maximum.

Finding the Local Minimum and Maximum

To find the local minimum and maximum of the function $f(x)=-2x^3+42x^2-270x+11$ , we need to find the critical points of the function. The critical points are the points where the derivative of the function is equal to zero or undefined. The derivative of the function $f(x)=-2x^3+42x^2-270x+11$ is given by:

f'(x) = -6x^2 + 84x - 270

To find the critical points, we need to solve the equation $f'(x) = 0$ . We can do this by setting the derivative equal to zero and solving for $x$ :

-6x^2 + 84x - 270 = 0

We can solve this quadratic equation using the quadratic formula:

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

In this case, $a = -6$ , $b = 84$ , and $c = -270$ . Plugging these values into the quadratic formula, we get:

x = \frac{-84 \pm \sqrt{84^2 - 4(-6)(-270)}}{2(-6)}

Simplifying this expression, we get:

x = \frac{-84 \pm \sqrt{7056 - 6480}}{-12}

x = \frac{-84 \pm \sqrt{576}}{-12}

x = \frac{-84 \pm 24}{-12}

This gives us two possible values for $x$ :

x = \frac{-84 + 24}{-12} = \frac{-60}{-12} = 5

x = \frac{-84 - 24}{-12} = \frac{-108}{-12} = 9

These are the critical points of the function $f(x)=-2x^3+42x^2-270x+11$ . To determine whether these points correspond to a local minimum or maximum, we need to examine the second derivative of the function.

Second Derivative

The second derivative of the function $f(x)=-2x^3+42x^2-270x+11$ is given by:

f''(x) = -12x + 84

We can use the second derivative to determine the nature of the critical points. If the second derivative is positive at a critical point, then the point corresponds to a local minimum. If the second derivative is negative at a critical point, then the point corresponds to a local maximum.

Evaluating the Second Derivative at the Critical Points

We can evaluate the second derivative at the critical points to determine the nature of the points:

f''(5) = -12(5) + 84 = -60 + 84 = 24

Since the second derivative is positive at $x = 5$ , this point corresponds to a local minimum.

f''(9) = -12(9) + 84 = -108 + 84 = -24

Since the second derivative is negative at $x = 9$ , this point corresponds to a local maximum.

Conclusion

In this article, we discussed the function $f(x)=-2x^3+42x^2-270x+11$ and its local minimum and maximum. We found that the function has one local minimum at $x = 5$ with value $f(5) = -2(5)^3 + 42(5)^2 - 270(5) + 11 = -2(125) + 42(25) - 270(5) + 11 = -250 + 1050 - 1350 + 11 = -539$ , and one local maximum at $x = 9$ with value $f(9) = -2(9)^3 + 42(9)^2 - 270(9) + 11 = -2(729) + 42(81) - 270(9) + 11 = -1458 + 3402 - 2430 + 11 = -1575$ .

Local Minimum and Maximum Values

Local Minimum/Maximum	Value
Local Minimum	$f(5) = -539$
Local Maximum	$f(9) = -1575$

Critical Points

Critical Point	Value
Local Minimum	$x = 5$
Local Maximum	$x = 9$

Second Derivative

Second Derivative	Value
$f''(5)$	$24$
$f''(9)$	$-24$

Conclusion

Introduction

In our previous article, we discussed the function $f(x)=-2x^3+42x^2-270x+11$ and its local minimum and maximum. We found that the function has one local minimum at $x = 5$ with value $f(5) = -539$ , and one local maximum at $x = 9$ with value $f(9) = -1575$ . In this article, we will answer some frequently asked questions about local minimum and maximum of a cubic polynomial.

Q: What is a local minimum?

A local minimum is a point on the graph of a function where the function value is smaller than or equal to the values at nearby points.

Q: What is a local maximum?

A local maximum is a point on the graph of a function where the function value is larger than or equal to the values at nearby points.

Q: How do I find the local minimum and maximum of a function?

To find the local minimum and maximum of a function, you need to find the critical points of the function. The critical points are the points where the derivative of the function is equal to zero or undefined. You can then use the second derivative to determine the nature of the critical points.

Q: What is the second derivative?

The second derivative of a function is the derivative of the first derivative of the function. It is used to determine the nature of the critical points of a function.

Q: How do I use the second derivative to determine the nature of the critical points?

If the second derivative is positive at a critical point, then the point corresponds to a local minimum. If the second derivative is negative at a critical point, then the point corresponds to a local maximum.

Q: Can a function have more than one local minimum or maximum?

Yes, a function can have more than one local minimum or maximum. In fact, a function can have multiple local minima and maxima.

Q: How do I find the local minimum and maximum of a function with multiple local minima and maxima?

To find the local minimum and maximum of a function with multiple local minima and maxima, you need to find all the critical points of the function and then use the second derivative to determine the nature of each critical point.

Q: Can a function have a local minimum or maximum at a point where the derivative is undefined?

Yes, a function can have a local minimum or maximum at a point where the derivative is undefined. This is known as a cusp or a corner point.

Q: How do I find the local minimum and maximum of a function with a cusp or corner point?

To find the local minimum and maximum of a function with a cusp or corner point, you need to examine the behavior of the function near the point where the derivative is undefined.

Q: Can a function have a local minimum or maximum at a point where the second derivative is zero?

Yes, a function can have a local minimum or maximum at a point where the second derivative is zero. This is known as a saddle point.

Q: How do I find the local minimum and maximum of a function with a saddle point?

To find the local minimum and maximum of a function with a saddle point, you need to examine the behavior of the function near the point where the second derivative is zero.

Conclusion

In this article, we answered some frequently asked questions about local minimum and maximum of a cubic polynomial. We hope that this article has been helpful in understanding the concept of local minimum and maximum of a function.

Frequently Asked Questions

Question	Answer
What is a local minimum?	A point on the graph of a function where the function value is smaller than or equal to the values at nearby points.
What is a local maximum?	A point on the graph of a function where the function value is larger than or equal to the values at nearby points.
How do I find the local minimum and maximum of a function?	Find the critical points of the function and then use the second derivative to determine the nature of each critical point.
What is the second derivative?	The derivative of the first derivative of a function.
How do I use the second derivative to determine the nature of the critical points?	If the second derivative is positive at a critical point, then the point corresponds to a local minimum. If the second derivative is negative at a critical point, then the point corresponds to a local maximum.
Can a function have more than one local minimum or maximum?	Yes, a function can have more than one local minimum or maximum.
How do I find the local minimum and maximum of a function with multiple local minima and maxima?	Find all the critical points of the function and then use the second derivative to determine the nature of each critical point.
Can a function have a local minimum or maximum at a point where the derivative is undefined?	Yes, a function can have a local minimum or maximum at a point where the derivative is undefined.
How do I find the local minimum and maximum of a function with a cusp or corner point?	Examine the behavior of the function near the point where the derivative is undefined.
Can a function have a local minimum or maximum at a point where the second derivative is zero?	Yes, a function can have a local minimum or maximum at a point where the second derivative is zero.
How do I find the local minimum and maximum of a function with a saddle point?	Examine the behavior of the function near the point where the second derivative is zero.