Question

Sketching curves Sketch a graph of a function \(f\) that is contimuous on \((-\infty, \infty)\) and has the following properties. $$f^{\prime}(x)>0, f^{\prime \prime}(x)>0$$

Short answer

Answer: The function f(x) is continuous, increasing, and concave up throughout its entire domain.

Step-by-step solution

Identify the function properties

We are given that the function \(f\) is continuous on \((-\infty, \infty)\). Also, its first derivative, \(f^{\prime}(x)\), is positive, which means that the function is increasing. Additionally, the second derivative, \(f^{\prime \prime}(x)\), is also positive, which implies the function is concave up.

Choose a suitable function

A suitable function that satisfies these properties is \(f(x) = x^2\) or any other quadratic function with a positive leading coefficient. In this case, we will use \(f(x) = x^2\) as our example to sketch.

Analyze the first derivative

We will analyze the first derivative, \(f^{\prime}(x)\), to understand the increase in the function. We will determine \(f^{\prime}(x)\) by differentiating \(f(x) = x^2\). $$f^{\prime}(x) = 2x$$ Since the first derivative, \(f^{\prime}(x) = 2x\), is positive for \(x>0\) and negative for \(x<0\), the function is increasing in its entire domain.

Analyze the second derivative

Now, we will analyze the second derivative, \(f^{\prime \prime}(x)\), to understand the concavity of the function. We will determine \(f^{\prime \prime}(x)\) by differentiating \(f^{\prime}(x)=2x\). $$f^{\prime \prime}(x) = 2 > 0$$ Since the second derivative, \(f^{\prime \prime}(x)\), is greater than zero throughout its entire domain, the function is concave up.

Sketch the graph

Finally, using the information obtained in previous steps, we will sketch the graph of \(f(x) = x^2\). Since the function is increasing and concave up on the entire domain, it will have a parabolic shape with a minimum point at the origin. The graph will start in the lower left quadrant, pass through the origin (minimum point), and then rise into the upper right quadrant.

Key concepts

Curve Sketching

When it comes to curve sketching, we are essentially drawing the shape of a graph based on the properties of its derivatives. This helps us understand the overall behavior of the function. Curve sketching involves analyzing the function’s increase or decrease and its curvature, which affects how the curve bends.
By understanding these properties, we can visualize a rough outline of what the graph should look like.
For the given function, we are told that:

• The first derivative, \(f'(x)\), is positive. This suggests the function is always increasing across its domain.
• The second derivative, \(f''(x)\), is also positive. This indicates the function is concave upward throughout.

Using a quadratic function, such as \(f(x) = x^2\), that satisfies these conditions, helps in illustrating these characteristics in a graph, depicting a smooth curve opening upwards.

Continuous Functions

Continuous functions are those that have no breaks, holes, or jumps across their domain. They are smooth curves that can be drawn without lifting a pencil. One key quality of a continuous function is that small changes in \(x\) result in small changes in \(f(x)\).
For our example with \(f(x) = x^2\), the function is continuous on \((-\infty, \infty)\), meaning there are no interruptions at any point on the graph.
Understanding the concept of continuous functions helps us when sketching curves, as it assures us that the graph will be an uninterrupted line or curve, consistent throughout its domain. This is crucial when attempting to visualize a function based on its derivative properties.

Derivatives analysis

Analyzing a function's derivatives helps us understand how it behaves in terms of growth, slope, and curvature. The first derivative, \(f'(x)\), provides insight into where the function increases or decreases; it acts like the slope of the tangent to the curve at any point. If \(f'(x) > 0\) always, the function increases everywhere.
The second derivative, \(f''(x)\), gives us information about the concavity of the function. If \(f''(x) > 0\), the function is concave up – the graph curves like a right-side-up bowl.
In our exercise, \(f'(x) = 2x\) suggests that \(f(x) = x^2\) is increasing for all \(x > 0\) and decreasing for \(x < 0\) based on the sign of \(f'(x)\). Similarly, \(f''(x) = 2\) tells us that the concavity is upward across the entire real line.

Concavity

Concavity refers to the direction the curve bends along its graph. The second derivative is the tool used to determine this property. When a function is concave up, it means the graph bends upwards like a cup, which is the case when \(f''(x) > 0\) throughout its domain.
For a function like \(f(x) = x^2\), the second derivative \(f''(x) = 2\) is positive, confirming that the graph is concave up everywhere.
Recognizing concavity also helps in identifying points of inflection, if any, which are the points where the graph changes from being concave up to concave down or vice versa. In our example, the entire graph is concave up, with no inflection points.