Problem 16 - Thomas Calculus

Question

In Exercises \(15-30\) , find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$ f(x)=-x-4, \quad-4 \leq x \leq 1 $$

Step-by-Step Solution

Answer

Absolute maximum is 0 at \((-4, 0)\), and absolute minimum is -5 at \((1, -5)\).

Step 1: Determine Critical Points

For the function \( f(x) = -x - 4 \), we first find the derivative, \( f'(x) \), to identify critical points. Since \( f(x) \) is a linear function, its derivative is \( f'(x) = -1 \). This derivative is constant and does not equal zero, indicating there are no critical points from the derivative.

Step 2: Evaluate Function at Endpoints

Since there are no critical points within the interval \([-4, 1]\), we evaluate the function at the endpoints of the interval. Evaluate \( f(-4) \) and \( f(1) \):- \( f(-4) = -(-4) - 4 = 4 - 4 = 0 \)- \( f(1) = -(1) - 4 = -1 - 4 = -5 \).

Step 3: Determine Absolute Extrema

Compare the function values at the endpoints to determine the absolute maximum and minimum. The values are \(0\) at \(x = -4\) and \(-5\) at \(x = 1\). - Absolute maximum: \(0\) at \((-4, 0)\)- Absolute minimum: \(-5\) at \((1, -5)\).

Step 4: Graph the Function and Identify Extrema

Graph the linear function \( f(x) = -x - 4 \) over the interval \([-4, 1]\). The line decreases from its maximum point at \((-4, 0)\) to its minimum point at \((1, -5)\). Mark these points on the graph, highlighting the absolute maximum and minimum.

Key Concepts

Critical Points

Critical points are where we look for potential peaks or valleys in a function. However, for the linear function described in our exercise, there aren't any such points from the derivative. This is because the derivative, which represents the function's slope, is constant. To find critical points, we usually set the derivative equal to zero and solve for x. But with the derivative here being a constant \( f'(x) = -1 \), it never reaches zero, indicating no critical points within our interval.

Interval Evaluation

Interval evaluation involves assessing the function's behavior at specific points on a given range. In our case, we're focused on the interval \([-4, 1]\). Since there are no critical points inside this interval due to the constant slope, we need to evaluate the function at the interval's endpoints to identify potential extrema. This means calculating the function values at \(x = -4\) and \(x = 1\). These computations help us identify where the absolute maximum and minimum values may occur on the closed interval.

Absolute Maximum and Minimum

Finding the absolute maximum and minimum values means identifying the highest and lowest points of the function within a specified range. From our interval evaluation, we calculate \( f(-4) = 0 \) and \( f(1) = -5 \). Comparing these values tells us that the maximum value is \(0\) at the point \((-4, 0)\), and the minimum value is \(-5\) at \((1, -5)\). These points illustrate the extremal behavior of the function over the interval.

Graphing Functions

When graphing a function, it's essential to capture its essential features, like the slope and extremal points. For the function \( f(x) = -x - 4 \), it represents a straight line because it is linear. The graph slopes downward, given the negative coefficient of \(x\). Plotting this line over the interval \([-4, 1]\), you observe it extends from the point \((-4, 0)\) and descends to \((1, -5)\). By marking these designated end points, we highlight where the maximum and minimum occur on the graph.

Derivative

Derivatives are fundamental in calculus for understanding how functions change. The derivative of a function gives us the slope of the tangent line at any given point. In our exercise, the derivative \( f'(x) = -1 \) informs us that the slope is consistently \(-1\). This indicates a straight, downward-sloping line. This negative constant derivative tells us there are no stationary points, also reinforcing that there are no critical points where the function might have a local maximum or minimum.