Question

Sketch the graph of a function with the given properties. \(f\) is continuous, but not necessarily differentiable, has domain \([0,6]\), reaches a maximum of 4 (attained when \(x=4)\), and a minimum of 2 (attained when \(x=2\) ). Additionally, \(f\) has no stationary points.

Short answer

Sketch a continuous, non-differentiable V-shaped graph with specified maxima at \(x=4\) and minima at \(x=2\).

Step-by-step solution

Understand the given properties

We need to sketch a function that is continuous on the domain \([0, 6]\), reaches a maximum value of 4 at \(x = 4\), a minimum value of 2 at \(x = 2\), and has no stationary points, which means there should be no places where the derivative is zero.

Plan the basic structure of the function

Since there are no stationary points, the function must keep increasing or decreasing without stopping. One simple way is to consider a piecewise linear function or use absolute values or constant changes in slope, such as creating a V-shape, that respect the given max/min points.

Sketch the draft of the function

Sketch a graph beginning from \((0,c)\) where \(c\) could be any number you choose to start with, decrease linearly to reach \( (2, 2) \) establishing the function's minimum, then increase linearly to reach \((4, 4)\) for the maximum, and finally decrease or increase again until \((6, d)\), where \(d\) is another number.

Verify continuity and non-differentiability

Ensure the graph doesn't jump or have holes (continuity), and that at each point where the slope changes, the transition is a sharp turn rather than smooth (non-differentiability), ensuring it fulfills all properties mentioned.

Key concepts

Graph Sketching

Graph sketching involves creating a visual representation of a function based on given traits. It's a skill that helps in understanding how a function behaves. When you sketch a graph, you translate mathematical properties into a picture. Think of it as capturing the essence of a function's behavior in a single glance.

For a function that is continuous but not differentiable, you can still sketch a meaningful graph by focusing on key points like where it achieves maximum or minimum values. In this exercise, for example, the function reaches a maximum of 4 at \(x = 4\) and a minimum of 2 at \(x = 2\). These are critical points that define the shape of the graph.

Try to maintain continuity, which means drawing the graph without lifting your pen off the paper. When graphing non-differentiable functions, use sharp corners or abrupt changes in direction to capture the places where the function doesn’t have a smooth tangent.

Piecewise Functions

Piecewise functions are functions defined by different expressions depending on the interval of the input. They offer great flexibility for modeling situations where behavior changes at specific points.

In our exercise, the absence of stationary points suggests a piecewise linear approach. This means the graph can be a combination of different linear segments, possibly changing direction quickly at the specified minimum and maximum points without having a smooth transition.

For instance, you might start with a linear segment from \(0, c\) to \(2, 2\) (where the minimum value is) and then switch to another linear segment to go up to \(4, 4\) (which is where the maximum is).

• From \(x=0\) to \(x=2\), the function decreases.
• From \(x=2\) to \(x=4\), it increases.
• Finally, from \(x=4\) to \(x=6\), it can either increase or decrease.

The segments are connected, ensuring continuity while having non-differentiable points where the slope changes sharply.

Non-differentiability

A function is non-differentiable at points where it does not have a unique tangent. This often occurs where there are sharp turns or cusps on the graph. Non-differentiability does not mean that a function is discontinuous. It is entirely possible for a function to be continuous everywhere yet non-differentiable at some points.

In this exercise, each change in direction on the graph of the function represents a non-differentiable point. Consider locations like \(x=2\) and \(x=4\) where the graph transitions sharply from one linear segment to another, providing examples of points of non-differentiability.

Non-differentiability is primarily about the slope, or derivative, not being defined at certain points, capturing the idea of 'no smooth turning' in the graph sketching plan. This creates interesting shapes that challenge intuitive notions of smoothness in continuous functions.

Domain and Range

The domain of a function is the set of input values (\(x\)-values) for which the function is defined, while the range is the set of output values (\(f(x)\)-values) that the function can take. Understanding domain and range is vital because they set the boundaries for possible values in a function.

For the function in this exercise, the domain is given as \[0, 6\]. This means the graph should only be drawn for \(x\) values starting from 0 up to 6. The function reaches a minimum of 2 and a maximum of 4, so these points help establish the range.

The range here will include all values between the minimum and maximum values, hence [2, 4]. Sketching within these bounds ensures that the function satisfies all given conditions, maintaining the correct behavior between defined minimum and maximum.