Which Interval For The Graphed Function Contains The Local Maximum?
Welcome to our blog post, where we will discuss the interval for the graphed function that contains the local maximum. This topic is important for students and professionals who are interested in mathematics, physics, engineering, or any other field that involves data analysis and visualization. In this article, we will provide you with some insights and tips for identifying the interval for the graphed function that contains the local maximum.
What is a Local Maximum?
Before we dive into the interval for the graphed function that contains the local maximum, let's define what a local maximum is. A local maximum is a point on a function where the function is highest in its immediate vicinity. In other words, it is a point that is higher than all the surrounding points, but it may not be the highest point on the entire function.
How to Identify a Local Maximum?
In order to identify a local maximum, we need to look at the graph of the function. We need to find a point where the slope of the function is zero or changes from positive to negative. This means that the function is increasing up to that point and then starts decreasing. The point at which this change occurs is the local maximum.
Let's take an example to understand this better. Suppose we have a function f(x) = x^3 - 6x^2 + 9x + 3. We can plot this function on a graph and look for the local maximum.
As we can see from the graph, the local maximum occurs at x = 2. This is the point where the slope of the function changes from positive to negative.
Interval for the Graphed Function that Contains the Local Maximum
Now that we know how to identify a local maximum, let's focus on the interval for the graphed function that contains the local maximum. The interval is the range of values of x where the local maximum occurs.
One way to find the interval is to look at the domain of the function. The domain is the set of all possible values of x that can be used as input for the function. In our example, the domain is all real numbers, since we can plug in any value of x and get a valid output.
Next, we need to find the critical points of the function. Critical points are the points where the slope of the function is zero or undefined. In our example, we can find the critical points by taking the derivative of the function f(x) and setting it equal to zero.
f'(x) = 3x^2 - 12x + 9 = 0
By solving this equation, we get x = 1 and x = 3. These are the critical points of the function.
Now, we need to check the sign of the second derivative of the function at each critical point. The second derivative tells us whether the critical point is a local maximum, a local minimum, or a point of inflection. If the second derivative is positive, the critical point is a local minimum. If it is negative, the critical point is a local maximum. If it is zero, the critical point is a point of inflection.
f''(x) = 6x - 12
At x = 1, f''(x) = -6, which means that it is a local maximum. At x = 3, f''(x) = 6, which means that it is a local minimum.
Therefore, the interval for the graphed function that contains the local maximum is (1, 2). This is the range of values of x where the function is increasing up to the local maximum at x = 2.
Conclusion
In conclusion, identifying the interval for the graphed function that contains the local maximum is an important skill for anyone who works with data and graphs. By understanding the concepts of local maximum, critical points, and second derivative, we can easily find the interval for the graphed function. Remember, the interval is the range of values of x where the local maximum occurs. We hope that this article has provided you with some useful insights and tips for identifying the interval for the graphed function that contains the local maximum. Happy graphing!
Remember to always practice and seek help when needed. Happy learning and exploring!
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