Problem 2 - Calculus with Concepts in Calculus

Question

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ f(x)=x^{2}+2 x+4 $$

Step-by-Step Solution

Answer

The function is concave upward on \( (-\infty, \infty) \) and concave downward nowhere.

Step 1: Understand the Second Derivative

For analyzing concavity, we need to determine the second derivative of the function. A function is concave upward where its second derivative is positive and concave downward where its second derivative is negative.

Step 2: Find the First Derivative

First, find the first derivative of the function \( f(x) = x^2 + 2x + 4 \). The first derivative \( f'(x) \) is obtained using basic differentiation rules:\[ f'(x) = 2x + 2 \]

Step 3: Find the Second Derivative

Differentiate the first derivative \( f'(x) = 2x + 2 \) to obtain the second derivative \( f''(x) \).\[ f''(x) = 2 \]

Step 4: Determine Concavity Intervals

Since the second derivative \( f''(x) = 2 \) is a constant and is positive for all \( x \), the function is concave upward for all real numbers. There are no intervals where the function is concave downward.

Key Concepts

Second Derivative

The second derivative of a function is a mathematical concept that helps determine a function's concavity. To find the second derivative, you first need to differentiate the given function to get its first derivative, and then differentiate the first derivative once more. In simpler terms, you are finding the rate of change of the rate of change. This can seem a bit abstract, so let's break it down. When you have a function, let's say, for instance, our function is given by \(f(x) = x^2 + 2x + 4\), finding the first derivative \(f'(x)\) simply involves taking the derivative as you usually would, resulting in \(f'(x) = 2x + 2\). Now, the second derivative \(f''(x)\) requires differentiating \(f'(x)\) once more, which in this case leads to \(f''(x) = 2\), a constant value. The significance of the second derivative lies in its ability to tell us about the concavity of the function. Specifically, it helps us understand how the slope of the function changes, providing insights into the curve’s shape. Let's explore these effects in more detail.

Concave Upward

A function is said to be concave upward on an interval where its second derivative is positive. This means that the graph of the function is curving upwards like a smile, even if it doesn't look exactly like a smile in the practical graph. In mathematical terms, wherever \(f''(x) > 0\), the curve is concave upwards. This is because a positive second derivative indicates that the slope of the tangent to the function (or the first derivative) is increasing. As a result, the line tangent to the curve rises as you move along the graph. For example, in our function \(f(x) = x^2 + 2x + 4\), we found that \(f''(x) = 2\) which is positive across all real numbers. Thus, the function is concave upward everywhere in its domain. You can think of it as a continuous upward slope, ensuring the function always turns upwards regardless of where you pick a point on it.

Concave Downward

In contrast, a function is concave downward on an interval when its second derivative is negative. This depicts a downward curve, similar to a frown, meaning that the function's slope decreases as you move along. Mathematically, when \(f''(x) < 0\), the curve is concave downward. Here, the tangent's slope decreases, indicating that the line tangent to the function falls as you move along the graph. This is the opposite of concave upward. For our provided function \(f(x) = x^2 + 2x + 4\), since \(f''(x) = 2\) is positive and never negative, there are no intervals where the function is concave downward. Thus, in this particular case, you will not encounter any downward turning in the curve across its entire slope. Understanding this distinction is key to analyzing the shape and behavior of functions on a graph.