Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.1.55

Chapter 4, Problem 4.1.55

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) = x+ cos⁻¹x on [-1,1]

Verified step by step guidance

First, understand that absolute extrema refer to the highest and lowest values of a function on a given interval. We need to find these values for ƒ(x) = x + cos⁻¹(x) on the interval [-1, 1].

To find the extrema, we should first check the endpoints of the interval. Evaluate ƒ(x) at x = -1 and x = 1.

Next, find the critical points within the interval by taking the derivative of ƒ(x) and setting it equal to zero. The derivative of ƒ(x) = x + cos⁻¹(x) is ƒ'(x) = 1 - (1/√(1-x²)).

Solve the equation ƒ'(x) = 0 to find the critical points. This involves solving 1 - (1/√(1-x²)) = 0, which simplifies to √(1-x²) = 1.

Evaluate ƒ(x) at the critical points found in the previous step. Compare these values with the values at the endpoints to determine the absolute maximum and minimum values on the interval [-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.

Critical Points

Critical points are values in the domain of a function where the derivative is either zero or does not exist. 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.

Endpoints of the Interval

Endpoints of the interval are the boundary values of the domain over which the function is being analyzed. In the context of finding absolute extrema, it is crucial to evaluate the function at these endpoints, as they can potentially yield the highest or lowest values. For the given function ƒ(x) = x + cos⁻¹(x) on the interval [-1, 1], the endpoints are x = -1 and x = 1.

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