Question

What does it mean for a function to have an absolute extreme value at a point \(c\) of an interval \([a, b] ?\)

Short answer

Short Answer: An absolute extreme value of a function over an interval is either a maximum or minimum value of the function within that interval. A continuous function has an absolute extreme value at a point c if it is a critical point (f'(c) = 0 or f'(c) doesn't exist) or it is an endpoint of the interval. For the example function f(x) = -x^2 + 4x in the interval [0, 4], there is an absolute maximum value of 4 at c=2 and absolute minimum values of 0 at both c=0 and c=4.

Step-by-step solution

Definition of Absolute Extreme Value

An absolute extreme value of a function \(f\) over an interval \([a, b]\) is a point within the interval \((c,f(c))\) such that the value of \(f(c)\) is either the largest (maximum) or smallest (minimum) compared to all other function values in the interval \([a, b]\). In other words, if a function has an absolute maximum value at point \(c\), then \(f(c) \geq f(x)\) for all \(x\) in the interval \([a, b]\). On the other hand, if a function has an absolute minimum value at point \(c\), then \(f(c) \leq f(x)\) for all \(x\) in the interval \([a, b]\).

Conditions for Absolute Extreme Value

A function can have an absolute extreme value at a point \(c\) under the following conditions: 1. \(f\) is continuous on the interval \([a, b]\). 2. \(c\) is either a critical point of the function (when \(f'(c) = 0\) or \(f'(c)\) does not exist) or an endpoint of the interval \([a, b]\). When these conditions are met, we can ensure that the function has an absolute extreme value at point \(c\).

Example of Absolute Extreme Value

Let's consider a continuous function \(f(x) = -x^2 + 4x\) in the interval \([0, 4]\). We will find the critical points and compare the function values at the critical points and the endpoints of the interval to find any absolute extreme values. 1. Calculate the first derivative of the function: \(f'(x) = -2x + 4\) 2. Find the critical points by setting \(f'(x) = 0\): \(-2x + 4 = 0 \Rightarrow x=2\) 3. Evaluate the function at the critical point and endpoints: \(f(0) = 0, f(2) = -4 + 8 = 4, f(4) = -16 + 16 = 0\) 4. Compare the function values: The maximum value is 4 and occurs at \(c=2\), while the minimum value is 0 and occurs at both \(c=0\) and \(c=4\). Thus, the function has an absolute maximum at \(c=2\) and absolute minimums at \(c=0\) and \(c=4\).

Key concepts

Critical Point

The concept of a critical point is pivotal when searching for extreme values of a function. A critical point occurs where the first derivative of a function is either zero or undefined. At these points, the function may change its trend from increasing to decreasing, or vice versa, which can indicate the presence of a local maximum or minimum value. To identify critical points, one typically calculates the first derivative of the function and sets it equal to zero. The solutions to this equation are potential critical points.

For instance, when examining the function in the exercise, by finding the first derivative and equating it to zero, we discovered that the point \(x=2\) is a critical point. It is essential to note that not all critical points will be absolute extrema, but they are initial candidates to check in the search for extreme values.

Continuous Function

A continuous function is one where small changes in the input \(x\) produce small changes in the output \(f(x)\). For the function to have an absolute extreme value on an interval, it must be continuous on that entire interval. Discontinuities, such as holes or jumps in the graph of the function, can prevent the function from having an absolute extreme value.

In our example, the quadratic function \(f(x) = -x^2 + 4x\) is continuous over the real number line, including the interval \[0, 4\]. This continuity ensures that we can properly evaluate the function at each point and compare its values to find the absolute maximum or minimum.

First Derivative

The first derivative of a function represents the slope of the tangent line at any given point on its graph. It tells us the rate at which the function value changes with respect to a change in the input. By setting the first derivative equal to zero, we can find points where the slope of the function is zero—these can be local maxima or minima. In our exercise, the first derivative is given by \(f'(x) = -2x + 4\). Calculating the first derivative is the first step in the process of finding extreme values and helps in determining the critical points of the function.

Function Interval Evaluation

To find the absolute extreme values of a function on an interval, the function values at all critical points and the endpoints of the interval must be evaluated and compared. This process is known as function interval evaluation. It’s essential because the absolute extreme values can occur at these critical points or the endpoints of the interval. In our given example, we evaluated the function at the critical point \(x=2\) and the endpoints \(x=0\) and \(x=4\). Upon comparison, the absolute maximum was determined to be at the critical point \(x=2\), and the absolute minimums were at the interval endpoints. This step of evaluating on the interval is crucial for identifying where the function reaches its highest and lowest values within the specified range.