Problem 83 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(f(t)\) has the property that \(f^{\prime}(a)=f^{\prime}(b)=0\) and that \(f(t)\) has two continuous derivatives. Use integration by parts to prove that \(\int_{a}^{b} f^{\prime \prime}(t) f(t) d t \leq 0\). Hint: Use integration by parts by differentiating \(f(t)\) and integrating \(f^{\prime \prime}(t)\). This result has many applications in the field of applied mathematics.

Step-by-Step Solution

Answer

The integral \( \int_a^b f''(t) f(t) \, dt \leq 0 \) is non-positive.

Step 1: Identify Functions for Integration by Parts

For integration by parts, we choose the functions as follows: Differentiating the function: \( u = f(t) \), hence \( du = f'(t) \, dt \). Integrating the function: \( dv = f''(t) \, dt \), hence \( v = f'(t) \) since the derivative of \( f'(t) \) gives \( f''(t) \).

Step 2: Apply Integration by Parts Formula

The integration by parts formula is given by \( \int u \, dv = uv - \int v \, du \). Substituting the identified functions, we have:\[ \int_a^b f''(t) f(t) \, dt = [f(t) f'(t)]_a^b - \int_a^b f'(t) f'(t) \, dt. \]

Step 3: Evaluate the Boundary Terms

The boundary terms \([f(t) f'(t)]_a^b\) are evaluated at the limits \(a\) and \(b\). Given that \(f'(a) = f'(b) = 0\), this term becomes:\[ [f(t) f'(t)]_a^b = f(b)f'(b) - f(a)f'(a) = 0 - 0 = 0. \]

Step 4: Simplify the Expression

The integral becomes:\[ \int_a^b f''(t) f(t) \, dt = -\int_a^b f'(t)^2 \, dt. \]Since the integrand \( f'(t)^2 \) is always non-negative (as squares are always non-negative), the right-hand side of the equation is less than or equal to zero.

Step 5: Conclude the Proof

Thus, we proved that:\[ \int_a^b f''(t) f(t) \, dt \leq 0. \]This integral is non-positive due to the fact that the integrand \( f'(t)^2 \) is non-negative.

Key Concepts

Second Derivative

The second derivative, denoted as \( f''(t) \), provides insight into the curvature or concavity of the function \( f(t) \). In simple terms, if \( f''(t) > 0 \) over an interval, the function is concave up (like a smile), and if \( f''(t) < 0 \), it is concave down (like a frown). This curvature indicates how the rate of change of \( f(t) \) itself is changing. In the context of integration by parts used in this problem, the second derivative \( f''(t) \) is integrated while the original function \( f(t) \) is differentiated. This dual approach helps to unravel complex integrals by transforming the problem into more manageable segments. Specifically, the behavior and properties of \( f''(t) \) are critical as they strongly influence the overall integral's value, leading to conclusions such as the integral being non-positive under certain conditions.

Boundary Terms

Boundary terms arise in the application of integration by parts. These terms, like \([f(t) f'(t)]_a^b\), are evaluated at the endpoints \(a\) and \(b\). They play a crucial role in simplifying integrals and often serve as hints about whether the original integral might simplify to zero or another specific value. In our problem, the conditions \( f'(a) = 0 \) and \( f'(b) = 0 \) simplify the boundary terms to zero. This drastically reduces the complexity of the integral. Understanding boundary terms helps identify other scenarios where functions have specific behaviors at endpoints, which frequently simplifies complicated calculations. In various applied mathematics problems, proper handling of these terms ensures accurate solutions that respect any given initial or boundary conditions.

Non-Positive Integral

An integral being non-positive implies that its value is either zero or negative. In this exercise, the integral \( \int_a^b f''(t) f(t) \ dt \) simplifies to \(-\int_a^b f'(t)^2 \ dt \). The integrand \( f'(t)^2 \) is always non-negative, as it's a square of another function. Hence, the entire integral becomes non-positive since we are effectively integrating the opposite of a non-negative quantity. Seeing it like this highlights a common characteristic of certain integrals in calculus, where a transformation leads to a non-positive conclusion, reflecting properties like energy conservation or stability in physics. This understanding informs us that under given conditions, the integral's value alerts us to system properties— such as indicating the absence of excess energy or indicating stability in modeling processes.

Proof Techniques

Proof techniques are vital in mathematics to validate statements or propositions. The integration by parts method is a classic technique for proving integral properties, such as the one seen here. By choosing specific functions to differentiate and integrate, we can strategically simplify and transform an apparently complex integral into simpler terms.

• Identify the appropriate functions: Determine which parts to differentiate and integrate.
• Apply the integration by parts formula: This step breaks down the integral into boundary terms and another integral.
• Evaluate the results at boundaries: This simplifies calculations using given conditions.
• Simplify and conclude: Ultimately, proving or disproving the problem statement.

Every step is crucial, serving as a toolset for mathematicians when dealing with constraints or seeking specific outcomes through proofs. This particular problem's steps are indicative of how systematic manipulation through integration by parts reveals deeper insights into function behavior.