Question

Show that $$ \begin{array}{l} \int_{-\infty}^{\infty}\left(t^{10}-t^{6}+5 t^{4}-5\right) e^{-t^{4}} d t \\ \leq \sqrt{\int_{-\infty}^{\infty}\left(t^{4}-1\right)^{2} e^{-t^{4}} d t} \sqrt{\int_{-\infty}^{\infty}\left(t^{6}+5\right)^{2} e^{-t^{4}} d t} \end{array} $$

Short answer

Using the Cauchy-Schwarz inequality, we can prove the inequality given in the exercise. The conditions of the inequality are met by the functions defined, so the inequality holds.

Step-by-step solution

Set up the Functions

First, let's identify the functions in the inequality given. We can set \(f(t) = (t^{4} - 1)\) and \(g(t) = (t^{6} + 5)\) and modify the given inequality as follows: \[\int_{-\infty}^{\infty} (fg)e^{-t^{4}} dt \leq \sqrt{\int_{-\infty}^{\infty} f^2 e^{-t^{4}} dt} \sqrt{\int_{-\infty}^{\infty} g^2 e^{-t^{4}} dt}\]

Apply Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality for integrals states that for any integrable functions \(f(t)\) and \(g(t)\) and for any interval [a, b], the following inequality holds: \[\left( \int_{a}^{b} |fg| dt \right)^2 \leq \int_{a}^{b} f^2 dt \int_{a}^{b} g^2 dt \] This inequality can be applied to our functions \(f(t)\) and \(g(t)\) because both are integrable, yielding the given inequality.

Conclude the Proof

Following the Cauchy-Schwarz Inequality, we have: \[ \left( \int_{-\infty}^{\infty} (fg)e^{-t^{4}} dt \right)^2 \leq \int_{-\infty}^{\infty} f^2 e^{-t^{4}} dt \int_{-\infty}^{\infty} g^2 e^{-t^{4}} dt \] which simplifies, following the functions we have defined and considering the square roots, to the inequality given in the exercise.

Key concepts

Mathematical Physics

Mathematical physics is a branch of physics that uses mathematical methods to solve physical problems. It's particularly involved in creating models to understand physical systems and can involve complex mathematical computations, including integrals over infinite limits and dealing with complex functions.

Within this context, the inequality in the exercise showcases an instance where mathematical physics and integral calculus intersect. Specifically, the exercise tackles evaluating an integral that arises within a physical context, which can be interpreted as calculating a certain property of a system, such as its energy or probability distributions, especially in quantum mechanics or statistical physics.

Applying inequalities like the Cauchy-Schwarz in mathematical physics provides bounds and limits within which the physical systems must operate. Such tools are crucial for theoretical analyses where exact solutions may be difficult to obtain or when assessing the stability and physical realities of the modeled systems.

Integral Calculus

Integral calculus is the study of the accumulation of quantities and the areas under and between curves. It's a fundamental part of mathematics, used widely in multiple fields such as physics, engineering, and economics.

The usage of integral calculus in the given exercise is an essential tool for evaluating the inequality, particularly because the integrals extend over the entire real line, from negative to positive infinity. This is a common scenario in physics for systems described throughout all space.

Improper integrals, such as those in the exercise, require careful consideration of the limits of integration and the behavior of the integrand at infinity. Integral calculus provides the methods by which we can rigorously evaluate or estimate these integrals, which in turn ensures the sound application of the Cauchy-Schwarz Inequality in proving the given statement.

Inequality Proof

An inequality proof in mathematics is a demonstration that a particular inequality holds between two expressions for all allowed values of the variables involved. Proofs involving inequalities are a staple in mathematics for establishing the relationships between different mathematical quantities.

In applying the Cauchy-Schwarz Inequality to the given exercise, we use a powerful method for proving that one mathematical expression is bounded by another. The technique used in the step-by-step solution applies this particular inequality within the framework of integral calculus and demonstrates how to validate that the original complex inequality indeed holds true.

Learning how to craft an inequality proof is critical for students, as it not only enhances understanding of the relationship between mathematical expressions but also develops rigorous logical reasoning and problem-solving skills, which are invaluable across all domains of mathematics.

Exponential Function

The exponential function, often denoted as \(e^x\), where \(e\) is the base of the natural logarithms, is crucial in both mathematics and its applications to natural sciences. The function grows or decays at a rate proportional to its value, a property which makes it germane to an array of phenomena like radioactive decay and population growth.

In the exercise, the exponential function appears as \(e^{-t^4}\), which serves as a weight function for the integrals involved. This is a characteristic feature in problems involving Gaussian integrals and thermodynamics where quantities are weighted by an exponential decay function, for instance, to model how certain properties diminish over time or space.

This highlights the importance of not only understanding the behavior of the exponential function due to its inherent properties but also its applicability in physical contexts, which requires familiarity with the way it modifies other mathematical objects, such as integrals in this case.