Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{5}-20 x+5\) on \([-2,0]\)

Step-by-Step Solution

Answer

The critical point is \(x = -1\), which is a local minimum. The function attains an absolute maximum of 25 at \(x = -1\) and an absolute minimum of -11 at \(x = 0\).

Step 1: Find the Derivative

The first step is to find the derivative of the function \(f(x)\). The derivative of the function \(f'(x)\) is \(5x^{4}-20\).

Step 2: Find the Critical Points

Next, set the derivative equal to zero or undefined to find critical points. The derivative is never undefined, so we only need to solve for when the derivative is equal to zero. Solving \(5x^{4} - 20 = 0\), the critical point on the interval \([-2, 0]\) is \(x = -1\).

Step 3: Classify the Critical Points

The next step is to determine if the critical point is a local minimum, local maximum, or neither. This can be made by evaluating the function \(f'(x)\) at points close to \(x = -1\). If \(f'(x) < 0\) to the left of \(x = -1\) and \(f'(x) > 0\) to the right of \(x = -1\), then \(x = -1\) is a local minimum. Conversely, if \(f'(x) > 0\) to the left of \(x = -1\) and \(f'(x) < 0\) to the right of \(x = -1\), then \(x = -1\) is a local maximum. In this case, \(x = -1\) is a local minimum.

Step 4: Find the Absolute Extrema

Lastly, to determine the absolute maximum and minimum on the interval \([-2, 0]\), we need to evaluate the function at the critical point and at the end points of the interval. The absolute maximum or minimum will be the highest or lowest of these values, respectively. Comparing the function values at \(x = -2, -1, 0\), we have that the function attains an absolute maximum of 25 at \(x = -1\) and an absolute minimum of -11 at \(x = 0\).

Key Concepts

Derivative of a Function

The derivative of a function is a foundational concept in calculus that measures how the function changes as its input changes. It's essentially the rate of change or the slope of the function at any given point. You can imagine it as the instantaneous speed of a car at any moment during a trip.

For the derivative of a polynomial function like the one given, which is of the form \(f(x) = x^5 - 20x + 5\), we apply the power rule. This rule states that to differentiate \(x^n\), you bring down the exponent as the coefficient of \(x\) and subtract one from the exponent. This leads to \(f'(x) = 5x^4 - 20\), where each term of the original polynomial function is dealt with individually. It's worth noting that differentiation is a process that can be applied to a wide variety of functions beyond polynomials, including exponential, logarithmic, trigonometric, and more complex combinations of functions.

Classifying Critical Points

Classifying critical points is a method used to determine the nature of specific points on the graph of a function where the derivative either equals zero or is undefined. At these points, the function may have a local extremum (either maximum or minimum) or a saddle point (neither a max nor a min).

In the context of the given function \(f(x) = x^5 - 20x + 5\), the critical point is found at \(x = -1\). To classify it, we analyze the sign of the derivative before and after the point. If the sign changes from positive to negative, the critical point is a local maximum, because the function was increasing and then starts to decrease. Conversely, if the sign of the derivative goes from negative to positive, as it does in this case, the critical point is a local minimum, indicating the function was decreasing and then began to increase. There are other tests like the Second Derivative Test that can help classify these points when the first derivative test is inconclusive. Understanding the behavior of the function around critical points is crucial in analyzing the graph and predicting its shape.

Absolute Extrema

The absolute extrema of a function on a closed interval is the highest or lowest point that the function reaches over that entire interval. This is in contrast to local extrema, which are the highest or lowest points in a small neighborhood around a point. To find absolute extrema, we often look at critical points and the boundaries of the given interval, since extrema can only occur at these places.

When analyzing \(f(x) = x^5 - 20x + 5\) on the interval \([-2,0]\), we compare the values of the function at the critical point (\(x = -1\)) and at the endpoints of the interval (\(x = -2\) and \(x = 0\)). The largest of these values is the absolute maximum, and the smallest is the absolute minimum. In the given exercise, the function achieves an absolute maximum value of 25 at \(x = -1\) and an absolute minimum value of -11 at \(x = 0\). This comparison is crucial for determining the overall highest and lowest points that the function will reach on the given interval, providing valuable information about the function's global behavior.