Problem 18 - 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. $$ f(x)=\frac{3}{4} x+2,[0,4] $$

Step-by-Step Solution

Answer

The absolute minimum of the function on the interval [0, 4] is 2 at \(x = 0\), and the absolute maximum is 5 at \(x = 4\).

Step 1: Compute the derivative

The derivative of the given function \( f(x) = \frac{3}{4}x + 2 \) is \( f'(x) = \frac{3}{4} \). This is because the derivative of \(x\) is \(1\) and the derivative of a constant is \(0\). So, the derivative of \(\frac{3}{4}x\) is \(\frac{3}{4}\), and the derivative of \(2\) is \(0\).

Step 2: Find the critical points

A critical point occurs where the derivative is either \(0\) or undefined. However, in this case, the derivative \( f'(x) = \frac{3}{4} \) is a constant which is never \(0\) or undefined, therefore, this function has no critical points.

Step 3: Evaluate the function at the endpoints

Since there are no critical points, the extrema can only occur at the endpoints of the interval. We therefore substitute the endpoints of the interval [0,4] into the original function to find the function values: \(f(0) = \frac{3}{4}(0) + 2 = 2\) and \(f(4) = \frac{3}{4}(4) + 2 = 5\).

Step 4: Identify the absolute extrema

By comparing the function values at the endpoints, it can be seen that the minimum value is \(2\) which occurs at \(x = 0\), and the maximum value is \(5\) which occurs at \(x = 4\). These are the absolute extrema of the function on the interval [0, 4].

Key Concepts

Critical Points in Calculus

When we talk about critical points in calculus, we're referring to places on the graph of a function where its derivative is zero or undefined. These points are essential in determining where a function may have local maxima or minima, which are the highest or lowest points within a certain interval.

In the context of our exercise, critical points would help us understand where the function changes direction or slope. However, as we found in the step-by-step solution, no critical points exist for the function because its derivative, which is a constant \(\frac{3}{4}\), does not have any points where it is zero or undefined.

This is a unique situation because usually, for a polynomial function or one with variable exponents, there would likely be points where the derivative equals zero. But for our linear function \(f(x) = \frac{3}{4}x + 2\), the constant slope implies a steady increase without any bends or turns where a local extremum could occur.

Absolute Extrema

Absolute extrema are the highest or lowest values that a function can take on a given interval. These can be either the tallest peaks or the deepest valleys in the graph of the function, and they occur at critical points or at the boundaries of the interval.

For our function, since there are no critical points, we focus on the interval boundaries to locate these extrema. As mentioned in the solution, we evaluated the function at the endpoints of the interval \([0,4]\). At \(x=0\), the function value is \(2\), and at \(x=4\), the function value is \(5\). These evaluations reveal that the absolute minimum value is \(2\), occurring at \(x=0\), and the absolute maximum value is \(5\), found at \(x=4\).

It's crucial to remember that the absolute extrema are the most extreme values the function attains within the specified closed interval, so they give us a complete picture of the function's range over that interval.

Derivative Computation

Derivative computation is a fundamental tool in calculus that deals with the rate at which a function changes. The derivative of a function at a point is the slope of the tangent line to the function's graph at that point. It's how you measure the instantaneous rate of change of the function with respect to one of its variables.

In the provided example, to find the derivative of the function \(f(x) = \frac{3}{4}x + 2\), we applied the basic derivative rules. The derivative of a constant multiple of \(x\), such as \(\frac{3}{4}x\), is just the constant \(\frac{3}{4}\). And since the derivative of a constant is zero, the derivative of \(2\) is \(0\). Thus, the complete derivative of \(f(x)\) is \(f'(x) = \frac{3}{4}\).

By mastering the computation of derivatives, you can not only find critical points but also understand the behavior of functions deeply, including their increasing or decreasing tendencies, and predict the shapes of their graphs.