Find The Absolute Extrema Of $f(x)$ On Each Of The Intervals.Given: $f(x) = (x-2) E^x$(a) Interval: 0 , 2 {0, 2} 0 , 2 (b) Interval: − 5 , 0 {-5, 0} − 5 , 0 - Write Answers In Terms Of E E E (e.g., E E E , $-

Mar 10, 2025 by ADMIN 213 views

Finding Absolute Extrema of a Function on Given Intervals

Introduction

In calculus, finding the absolute extrema of a function on a given interval is a crucial concept. It involves determining the maximum and minimum values of the function within the specified interval. In this article, we will explore how to find the absolute extrema of the function $f(x) = (x-2) e^x$ on two different intervals: [0, 2] and [-5, 0].

Understanding the Function

Before we proceed, let's understand the given function $f(x) = (x-2) e^x$. This function is a product of two functions: a linear function $(x-2)$ and an exponential function $e^x$. The exponential function $e^x$ is a continuously increasing function, and the linear function $(x-2)$ is a linear function with a slope of 1 and a y-intercept of -2.

Finding Critical Points

To find the absolute extrema of the function, we need to find the critical points. Critical points are the points where the function changes from increasing to decreasing or vice versa. To find the critical points, we need to find the derivative of the function and set it equal to zero.

Derivative of the Function

To find the derivative of the function $f(x) = (x-2) e^x$, we will use the product rule of differentiation. The product rule states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.

Applying the product rule to the given function, we get:

f'(x) = (x-2) e^x + e^x

Simplifying the derivative, we get:

f'(x) = e^x (x-2 + 1)

f'(x) = e^x (x-1)

Setting the Derivative Equal to Zero

To find the critical points, we need to set the derivative equal to zero and solve for x.

e^x (x-1) = 0

Since $e^x$ is never equal to zero, we can divide both sides by $e^x$.

x-1 = 0

Solving for x, we get:

x = 1

Finding the Absolute Extrema on the Interval [0, 2]

Now that we have found the critical point, we need to determine whether it is a maximum or a minimum. To do this, we need to examine the behavior of the function on either side of the critical point.

Since the critical point x = 1 is within the interval [0, 2], we need to examine the behavior of the function on the intervals [0, 1] and [1, 2].

On the interval [0, 1], the function is increasing, and on the interval [1, 2], the function is decreasing. This indicates that the critical point x = 1 is a maximum.

Finding the Absolute Extrema on the Interval [-5, 0]

Now that we have found the critical point, we need to determine whether it is a maximum or a minimum. To do this, we need to examine the behavior of the function on either side of the critical point.

Since the critical point x = 1 is not within the interval [-5, 0], we need to examine the behavior of the function on the intervals [-5, 1] and [1, 0].

On the interval [-5, 1], the function is increasing, and on the interval [1, 0], the function is decreasing. This indicates that the critical point x = 1 is a maximum.

Conclusion

In conclusion, we have found the absolute extrema of the function $f(x) = (x-2) e^x$ on the intervals [0, 2] and [-5, 0]. The critical point x = 1 is a maximum on both intervals.

Final Answer

The absolute extrema of the function $f(x) = (x-2) e^x$ on the interval [0, 2] is $f(1) = (1-2) e^1 = -e$.

The absolute extrema of the function $f(x) = (x-2) e^x$ on the interval [-5, 0] is $f(1) = (1-2) e^1 = -e$.

References

Calculus by Michael Spivak
Calculus by James Stewart

Introduction

In our previous article, we explored how to find the absolute extrema of the function $f(x) = (x-2) e^x$ on two different intervals: [0, 2] and [-5, 0]. In this article, we will answer some frequently asked questions related to finding absolute extrema of a function on given intervals.

Q&A

Q: What is the difference between a local maximum and a global maximum?

A: A local maximum is a point where the function changes from increasing to decreasing, while a global maximum is the highest point on the entire function.

Q: How do I find the absolute extrema of a function on a given interval?

A: To find the absolute extrema of a function on a given interval, you need to find the critical points by setting the derivative of the function equal to zero and solving for x. Then, you need to examine the behavior of the function on either side of the critical point to determine whether it is a maximum or a minimum.

Q: What is the significance of the derivative in finding absolute extrema?

A: The derivative of a function is used to find the critical points, which are the points where the function changes from increasing to decreasing or vice versa. The derivative is also used to determine the behavior of the function on either side of the critical point.

Q: Can a function have multiple absolute extrema on a given interval?

A: Yes, a function can have multiple absolute extrema on a given interval. This occurs when the function has multiple critical points within the interval.

Q: How do I determine whether a critical point is a maximum or a minimum?

A: To determine whether a critical point is a maximum or a minimum, you need to examine the behavior of the function on either side of the critical point. If the function is increasing on one side and decreasing on the other, then the critical point is a maximum. If the function is decreasing on one side and increasing on the other, then the critical point is a minimum.

Q: What is the role of the second derivative in finding absolute extrema?

A: The second derivative of a function is used to determine the concavity of the function. If the second derivative is positive, then the function is concave up, and if the second derivative is negative, then the function is concave down. This information can be used to determine whether a critical point is a maximum or a minimum.

Q: Can a function have no absolute extrema on a given interval?

A: Yes, a function can have no absolute extrema on a given interval. This occurs when the function is either increasing or decreasing throughout the entire interval.