Problem 69 - Calculus of a Single Variable

Question

Sketching a Graph Sketch the graph of the arbitrary function \(f\) such that $$ f^{\prime}(x)\left\\{\begin{array}{ll}{ > 0,} & {x < 4} \\ {\text { undefined, }} & {x=4} \\ { < 0,} & {x > 4}\end{array}\right. $$

Step-by-Step Solution

Answer

The function is increasing from negative infinity to \(x=4\), reaches a peak at \(x=4\), and then decreases from \(x=4\) to positive infinity.

Step 1: Plot the intervals of increase and decrease

On a number line, mark the interval where the function is increasing or decreasing according to the derivative. The function increases on (-Infinity, 4) and decreases on (4, Infinity).

Step 2: Identify point of change for the derivative

The derivative changes at \(x=4\), where it goes from positive to negative, indicating a maximum or minimum point or a point of inflection. However, as the function changes from increasing to decreasing at \(x=4\), this point will be a local maximum.

Step 3: Sketch the graph

Draw the y-axis and x-axis on a graph. Sketch a curve that increases from negative infinity up to \(x=4\), reaches a peak at \(x=4\), and then decreases from \(x=4\) to positive infinity. It creates a smooth curve that looks like a hill.

Key Concepts

Derivative Function

Understanding the derivative of a function is crucial for graph analysis in calculus. The derivative, represented by f'(x), is the rate at which the function's output value changes as its input value changes. In simple terms, it tells you how steep the graph of the function is at any given point.

If the derivative is positive, the function is increasing; if it's negative, the function is decreasing. When the derivative is zero, the graph may have a horizontal tangent line, often indicating a local maximum or minimum. However, the derivative being undefined at a certain point, like in our exercise at x=4, suggests the presence of a cusp, corner or vertical tangent, which could also be the location of a local maximum or minimum.

In the exercise, we use this information to determine how the graph behaves around x=4. Since the derivative changes from positive to negative around this point, it tells us the graph of the function peaks there, leading us to conclude that x=4 is likely the location of a local maximum.

Intervals of Increase and Decrease

Graphing a function's behavior over intervals can reveal its overall structure. Based on the derivative's sign, we categorize intervals where the function increases or decreases. A positive derivative, as noted, indicates that the function is increasing, and conversely, a negative derivative indicates a decrease.

In our exercise, f'(x) > 0 when x < 4, marking that interval as the region where the function's graph is rising. For values where x > 4, the negative derivative, f'(x) < 0, means the function's graph is descending. When sketching the graph, these intervals are crucial as they dictate the slope of the curve before and after the point where the derivative is undefined (our suspected local maximum).

Local Maximum

A local maximum is a point on the graph where the function value is higher than at any neighboring points within a certain range. It's like the peak of a hill. To determine a local maximum, we rely on both the intervals of increase and decrease and the derivative's behavior.

In our case, as the derivative transitions from positive to negative at x=4, there is a switch from an increasing interval to a decreasing interval. This pivotal change in the derivative's sign is a strong indication of a local maximum. It is visually represented in the graph as the highest point on the curve within its local vicinity, essentially the 'tip' of the hill. To confirm this further, we should verify if the function's second derivative is negative at this point, but this is beyond the scope of our current exercise.