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.

Short answer

\(\int_{a}^{b} f''(t) f(t) dt \leq 0 \) because \(\int_{a}^{b} f'(t)^2 dt \geq 0\).

Step-by-step solution

Set up the integration by parts formula

The integration by parts formula is given by \( \int u \cdot v' \, dt = uv - \int v \cdot u' \, dt \). Here, choose \( u = f(t) \) and \( v' = f''(t) \). This means \( u' = f'(t) \) and \( v = f'(t) \).

Apply integration by parts

Substitute the expressions for \( u \), \( v \), and their derivatives into the formula: \[ \int_{a}^{b} f(t) f''(t) \, dt = \left[ f(t)f'(t) \right]_{a}^{b} - \int_{a}^{b} f'(t)^2 \, dt. \]

Evaluate the boundary terms

Evaluate \( \left[ f(t)f'(t) \right]_{a}^{b} = f(b)f'(b) - f(a)f'(a) \). Since \( f'(a) = 0 \) and \( f'(b) = 0 \), this expression simplifies to \( 0 - 0 = 0 \).

Simplify the integral

The expression becomes: \[ \int_{a}^{b} f(t) f''(t) \, dt = - \int_{a}^{b} f'(t)^2 \, dt. \]

Analyze the result for negativity

Since \( f'(t)^2 \) is always non-negative, \( -\int_{a}^{b} f'(t)^2 \, dt \leq 0 \) because the integral of a non-negative function is non-negative, thus negating it produces a non-positive result.

Key concepts

Continuous Derivatives

Continuous derivatives are derivatives that exist and are smooth, meaning they don’t have any abrupt changes, over the entire domain of a function. In simpler terms, if the derivative of a function doesn’t "jump" or "break" at any point, it is continuous. For the given problem, where the function \( f(t) \) has two continuous derivatives, it ensures a smooth behavior for both \( f'(t) \) and \( f''(t) \). This smoothness is crucial when applying techniques like integration by parts. It guarantees that integration can be carried out smoothly without encountering any discontinuities.

**Why is this important?**

• Ensures no undefined behavior.
• Validates the use of calculus operations without issues.
• Enables accurate estimation or modeling of real-world phenomena.

Uninterrupted smoothness of derivatives also facilitates easier application of mathematical methods to solve equations, making it highly valued in applied mathematics.

Boundary Conditions

Boundary conditions are specific requirements or constraints placed on the values of a function at the edges of its domain. In the exercise, the boundary conditions are that the derivative \( f'(t) \) equals zero at the points \( a \) and \( b \). This means that at these "boundaries," the rate of change of the function is zero, pointing to critical points such as a peak, valley, or inflection point.

**Significance in Integration by Parts:**

• They allow the boundary terms to cancel out simplifying calculations.
• Turn the integration limits into zero, which leads to a simpler expression.

In the proof, these boundary conditions dramatically simplify the integration by parts result by turning the expression \([f(t)f'(t)]_{a}^{b}\) into zero. Thus, boundary conditions can greatly influence the outcomes of function evaluations and transformations.

Applied Mathematics

Applied mathematics uses mathematical techniques and principles to solve real-world problems, which is why it’s deeply intertwined with fields like physics, engineering, and economics. In the provided exercise, integration by parts showing \( \int_{a}^{b} f''(t) f(t) dt \leq 0 \) is vital within numerous applied mathematics scenarios. For instance, energy minimization problems, stability analysis, and more.

**Integration by Parts in Applied Settings:**

• Can model physical systems and derive equations for various phenomena.
• Helps verify stability and feasibility within dynamic systems.
• Assists in deducing optimal conditions for system performance.

Through this exercise, one learns to harness the rigorous mathematics to bridge theoretical results with practical applications, making integration by parts not just a theoretical tool but a powerful component in applied mathematics.