Problem 51 - Calculus Single Variable

Question

Suppose that \(f(x)>0\) and \(f^{\prime \prime}(x)>0\) for all \(x\). Describe the concavity of \(f^{2}\).

Step-by-Step Solution

Answer

The function \(f^2(x)\) is concave up for all \(x\).

Step 1: Differentiate f²(x)

To analyze the concavity of the function \(f^2(x)\), first find its first derivative. Let \(g(x) = f^2(x)\). Then, the derivative of \(g(x)\) is given by applying the chain rule:\[g'(x) = 2f(x)f'(x)\]

Step 2: Differentiate g'(x) to find g''(x)

To determine concavity, we need the second derivative \(g''(x)\). Differentiate \(g'(x) = 2f(x)f'(x)\) using the product rule:\[g''(x) = 2[f'(x)f'(x) + f(x)f''(x)] = 2[f'(x)^2 + f(x)f''(x)]\]

Step 3: Analyze the sign of g''(x)

The concavity of the function \(g(x) = f^2(x)\) is determined by the sign of \(g''(x)\). Given \(f(x) > 0\) and \(f''(x) > 0\), both terms \(f'(x)^2\) and \(f(x)f''(x)\) in \(g''(x)= 2[f'(x)^2 + f(x)f''(x)]\) are positive. Therefore, \(g''(x) > 0\) for all \(x\).

Step 4: Conclude the concavity

Because \(g''(x) > 0\) for all \(x\), the function \(g(x) = f^2(x)\) is concave up everywhere. Concavity up indicates that the function is always curving upwards, resembling a U-shape. Therefore, \(f^2(x)\) is concave up for all \(x\).

Key Concepts

Second Derivative

The second derivative gives us vital information about the concavity of a function. When we have a function, let's call it \(g(x)\), its first derivative \(g'(x)\) tells us the rate at which \(g(x)\) is changing. However, to understand how \(g(x)\) is curving, we need to find its second derivative, written as \(g''(x)\). If \(g''(x) > 0\) for all \(x\), the function is concave up. This means the graph is curving upwards, similar to a U-shape. Conversely, if \(g''(x) < 0\), the function is concave down, like an inverted U-shape.

For the function \(f^2(x)\), as derived in the steps, \(g''(x) = 2[f'(x)^2 + f(x)f''(x)]\). With both terms \(f'(x)^2\) and \(f(x)f''(x)\) being positive due to given conditions, \(g''(x)\) remains greater than zero across all \(x\), confirming its concavity upwards. Using the second derivative is crucial in calculus for determining these kinds of graphical properties.

Chain Rule

The chain rule is an indispensable tool in calculus used for differentiating composite functions. A composite function is a function made up of two or more functions. If you have, for instance, a function \(h(x) = f(g(x))\), the chain rule helps you find the derivative \(h'(x)\) by differentiating the outer function \(f\) and the inner function \(g\).

In the context of \(g(x) = f^2(x)\), we apply the chain rule to find \(g'(x)\). Here, the function \(f^2(x)\) can be viewed as \((f(x))^2\), where \(f(x)\) is an inner function, and squaring is the outer function. By applying the chain rule, we differentiate the outer function first and then multiply by the derivative of the inner function \(f(x)\). This gives us:\[g'(x) = 2f(x)f'(x)\].

The chain rule simplifies the differentiation process of complex functions into manageable steps, making it simple to handle problems involving layered functions.

Product Rule

When differentiating products of two functions, such as \(u(x)\) and \(v(x)\), the product rule is ideal. Multiplying functions is common in calculus, and the product rule provides a reliable method for finding their derivatives. The product rule states that if \(h(x) = u(x)v(x)\), then the derivative \(h'(x)\) is given by

• \(h'(x) = u'(x)v(x) + u(x)v'(x)\)

In simple terms, you differentiate one function, keep the other function unchanged, add them together but then switch the roles. For our function \(g(x) = 2f(x)f'(x)\) derived from \(f^2(x)\), this means applying the product rule to \(2f(x)f'(x)\) during the second differentiation step:

\[g''(x) = 2[f'(x)f'(x) + f(x)f''(x)] = 2[f'(x)^2 + f(x)f''(x)]\].
With practice, the product rule becomes intuitive and plays a crucial role when working with multiplying functions.

Derivatives in Calculus

Derivatives lie at the heart of calculus, serving as the foundation for various mathematical concepts, including concavity, optimization, and motion. A derivative represents the rate at which a quantity changes. If we think of a graph as a landscape, derivatives allow us to understand its slopes, peaks, and troughs.

When tackling problems, understanding how to calculate derivatives, whether through basic rules, the chain rule, or the product rule, is crucial. Each rule has its right place, such as:

• The basic power rule for straightforward polynomials.
• The chain rule for nested functions or compositions.
• The product rule for multiplicative functions.

Combining these tools effectively allows us to solve complex calculus problems, analyze graphs, and understand how functions behave. By applying these derivative rules systematically, calculus becomes not only manageable but insightful, offering a clear view into the workings of mathematical relationships.