Textbook Question
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x sin⁻¹ x on [-1, 1]
5. Graphical Applications of Derivatives
Finding Global Extrema
6:26 minutesProblem 4.1.53
Textbook Question
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = (2x)ˣ on [0.1,1]
Verified step by step guidance1
Step 1: Understand the problem. We need to find the absolute maximum and minimum values of the function ƒ(x) = (2x)^x on the interval [0.1, 1]. This involves evaluating the function at critical points and endpoints within the interval.
Step 2: Find the derivative of the function ƒ(x) = (2x)^x. To do this, first express the function in a form that is easier to differentiate: ƒ(x) = e^(x ln(2x)). Use the chain rule and the product rule to differentiate this expression.
Step 3: Set the derivative equal to zero to find the critical points. Solve the equation derived from the derivative to find the values of x where the derivative is zero. These are the critical points where the function could have local maxima or minima.
Step 4: Evaluate the function ƒ(x) at the critical points found in Step 3, as well as at the endpoints of the interval, x = 0.1 and x = 1. This will help determine the absolute maximum and minimum values of the function on the interval.
Step 5: Compare the values of ƒ(x) at the critical points and endpoints. The largest value will be the absolute maximum, and the smallest value will be the absolute minimum on the interval [0.1, 1].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Extrema
Absolute extrema refer to the highest and lowest values of a function over a specified interval. To find these values, one must evaluate the function at critical points, where the derivative is zero or undefined, as well as at the endpoints of the interval. The largest of these values is the absolute maximum, while the smallest is the absolute minimum.
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Critical Points
Critical points are values in the domain of a function where its derivative is either zero or undefined. These points are essential for finding absolute extrema, as they indicate where the function may change direction. To locate critical points, one must first compute the derivative of the function and solve for when it equals zero or is undefined.
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Evaluating Functions on an Interval
Evaluating functions on a closed interval involves determining the function's values at both endpoints and any critical points within the interval. This process is crucial for identifying absolute extrema, as it ensures that all potential maximum and minimum values are considered. The interval's endpoints and critical points collectively provide a complete picture of the function's behavior within that range.
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