Problem 30 - Calculus of a Single Variable

Question

Finding Extrema on a Closed Interval In Exercises \(17-36\) , find the absolute extrema of the function on the closed interval. $$ g(x)=|x+4|,[-7,1] $$

Step-by-Step Solution

Answer

The absolute maximum of the function \(g(x)=|x+4|\) on the interval \([-7,1]\) is 5 at \(x = 1\), and the absolute minimum is 0 at \(x = -4\).

Step 1: Understanding the Function

First, we need to understand the function \(g(x)=|x+4|\). This function is the absolute value of \(x+4\), which means it will always be positive or zero. It reaches its minimum when \(x+4\) is equal to 0.

Step 2: Finding the Critical Points

The critical points of a function are the points where the function changes direction or is not differentiable. For the absolute value function, this happens when the input is 0. Setting \(x+4 = 0\), we find \(x = -4\). This is within our interval of \([-7,1]\), so it will be a critical point.

Step 3: Evaluating the Function at the Ends of the Interval and at the Critical Point

We have to find the values of the function at the ends of the interval and at the critical point to find the absolute extrema. We find \(g(-7)=|-7+4|=3\), \(g(-4)=|-4+4|=0\), and \(g(1)=|1+4|=5\).

Step 4: Comparing the Values to Find the Absolute Extrema

From our evaluations in the previous step, we see that the maximum value of the function on this interval is 5 and the minimum value is 0. So, the absolute maximum is 5 at \(x = 1\), and the absolute minimum is 0 at \(x = -4\).

Key Concepts

Closed Interval

A closed interval is a range of values that includes its endpoints. In the context of the exercise, we have the interval \([-7, 1]\), which means both -7 and 1 are included in the set of possible values for \(x\). Closed intervals are significant when finding absolute extrema because:

• They define the range within which we are interested in finding the extrema.
• Including the endpoints means the extrema could exist at the boundary of the interval.

When evaluating functions over a closed interval, we must check the function's values at both the boundaries and any critical points within. This ensures that we find the highest and lowest values in the entire interval, accounting for all possibilities.

Critical Points

Critical points of a function are the specific values of the input where the derivative is either zero or undefined. These points are important because they can indicate where the function switches from increasing to decreasing or vice versa. To find critical points of \(g(x) = |x+4|\), consider:

• The absolute value function \(|x+4|\) makes a sharp turn at \(x=-4\), because the expression inside the absolute value equates to zero.
• This point \(-4\) is where the function is not differentiable, making it a critical point.

Because \(-4\) is within the closed interval \([-7, 1]\), it is especially important to consider it when finding extrema.

Absolute Value Function

The absolute value function takes any real number and returns its non-negative magnitude. It is written as \(|x|\), and behaves like this:

• If the input is positive or zero, its absolute value is the number itself.
• If the input is negative, its absolute value is the number without the negative sign.

For \(g(x) = |x+4|\), this function translates the graph of the linear function \(x+4\) into one that is symmetrical about the y-axis. The absolute value function is not differentiable wherever the inside expression equals zero, highlighting the critical nature of that point. This property of turning negative values to positive ones significantly impacts the calculation of extrema, especially in the range provided.

Differentiability

Differentiability refers to whether a function has a derivative at a particular point. A function that is differentiable throughout an interval has a smooth curve without breaks, corners, or cusps. However, the absolute value function, like \(g(x) = |x+4|\), has a sharp point at \(x=-4\), making it non-differentiable at that exact point. Here's why differentiability matters:

• It is necessary to determine where a function may have critical points.
• If a function is not differentiable at a point, it can still have a critical point there due to the change in direction of the graph.

In exercises dealing with finding absolute extrema, distinguishing between differentiable and non-differentiable points is vital, especially when working with absolute value functions on closed intervals.