Question

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(0)=0, f(1)=2\); (c) \(f\) is an even function; (d) \(f^{\prime}(x)>0\) for \(x>0\); (e) \(f^{\prime \prime}(x)>0\) for \(x>0\).

Short answer

Plot points (0,0), (1,2), (-1,2); ensure symmetry and upward curvature.

Step-by-step solution

Understanding the Properties

We need to sketch a graph of a function that meets the given criteria: continuous everywhere, passes through specific points, is even, and has first and second derivatives greater than 0 for positive x.

Graph Properties of an Even Function

Since function \(f\) is even, for every point \((x, f(x))\) on the graph, the point \((-x, f(x))\) must also be on the graph. This means the graph is symmetric about the y-axis.

Determining Key Points and Slope

We know \(f(0) = 0\) and \(f(1) = 2\). Since \(f\) is even, \(f(-1) = 2\) as well. The derivative \(f^{\prime} (x) > 0\) for \(x > 0\) indicates the function is increasing on positive x.

Determining Concavity

Since \(f^{\prime \prime} (x) > 0\) for \(x > 0\), the function is concave up on positive x. This implies the rate of increase is itself increasing for positive x.

Sketch the Graph

Start by plotting the points \((0, 0)\), \((1, 2)\), and \((-1, 2)\). The function should be symmetric around the y-axis. Since the function is increasing and concave up for positive x, sketch an upward curving line between \((0, 0)\) and \((1, 2)\), and symmetrically for \((-1, 2)\). Extend this shape smoothly to cover the entire graph, maintaining the symmetry and continuity.

Key concepts

Understanding Continuous Functions

Continuous functions are mathematical functions that have no breaks, jumps, or holes in their graphs. This means that the graph of a continuous function can be drawn without lifting your pencil from the paper. In formal terms, a function \( f(x) \) is continuous at a point \( x = c \) if the limit as \( x \) approaches \( c \) exists and is equal to the function's value at that point: \[ \lim_{x \to c} f(x) = f(c) \]. This property helps ensure that all points on the graph are smoothly connected. For an even function like \( f \), being continuous everywhere also involves being continuous at every point across the symmetric curve. This type of symmetry guarantees that the left side of the graph mirrors the right, maintaining continuity throughout.

Exploring the First Derivative

The first derivative, \( f^{\prime}(x) \), provides insights into the rate at which the function \( f(x) \) is increasing or decreasing. If \( f^{\prime}(x) > 0 \) for \( x > 0 \), it implies that the function is increasing over this interval. This tells us that as \( x \) gets larger, so does \( f(x) \). In our case, where \( f \) is an even function, the symmetry means that the graph also mirrors this increasing behavior in a decreasing fashion on the negative side of the y-axis. Understanding the behavior of the first derivative helps in predicting the function’s shape and direction as it progresses.

Examining the Second Derivative

The second derivative, \( f^{\prime\prime}(x) \), reveals information about the concavity of the function. When \( f^{\prime\prime}(x) > 0 \) for \( x > 0 \), this suggests the graph of the function is concave up. This means that not only is the function increasing, but the rate at which it increases is also accelerating. For an even function, this concave-up behavior on the positive x-axis should symmetrically apply on the negative x-axis as well. This property shapes the graph to appear as a bowl opening upwards, indicating the function's curvature and aiding with sketching how it progresses over different regions.

Tips for Graph Sketching

When sketching the graph of a function, it's essential to consider various elements to ensure accuracy. Start by plotting critical given points. In this exercise, these points are \((0, 0)\), \((1, 2)\), and by symmetry, \((-1, 2)\). Next, use symmetry about the y-axis because the function is even, meaning the left and right sides of the graph should mirror each other. With the first derivative being positive, ensure the function is rising as it moves away from the y-axis on the positive side. The second derivative being positive tells you to shape the graph to curve upwards, hinting at a steeper rise as x-value becomes larger. Overall, consider connecting these points smoothly and symmetrically to illustrate the continuous nature of the function. Finally, ensure the graph reflects both increasing and concave-up characteristics as dictated by the derivatives.