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).
ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]
5. Graphical Applications of Derivatives
Finding Global Extrema
3:09 minutesProblem 4.1.51
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) = sin 3x on [-π/4,π/3]
Verified step by step guidance1
First, identify the critical points of the function ƒ(x) = sin(3x) within the interval [-π/4, π/3]. To do this, find the derivative of ƒ(x), which is ƒ'(x) = 3cos(3x).
Set the derivative ƒ'(x) = 3cos(3x) equal to zero to find the critical points: 3cos(3x) = 0. Solve for x to find the critical points within the interval.
Evaluate the function ƒ(x) = sin(3x) at the critical points found in the previous step, as well as at the endpoints of the interval, x = -π/4 and x = π/3.
Compare the values of ƒ(x) at the critical points and the endpoints to determine which is the absolute maximum and which is the absolute minimum.
Conclude by stating the location and value of the absolute maximum and minimum of ƒ(x) on the interval [-π/4, π/3].
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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 of the independent variable where the derivative of the function is either zero or does not exist. These points are essential in determining the behavior of the function, as they can indicate potential locations for local maxima or minima. To find 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 a function on a closed interval involves checking the function's values at both endpoints and at any critical points found within the interval. This process ensures that all potential candidates for absolute extrema are considered. The function's behavior can vary significantly across the interval, making this evaluation crucial for accurately identifying absolute maximum and minimum values.
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