A particle moves along a coordinate line, its position at...

A particle moves along a coordinate line, its position at time t given by the function $x(t)=A e^{k \gamma}+B e^{-k t} . \quad(A>0 . B>0, k>0)$ (a) Find the times $t$ at which the particle is closest to the origin. (b) Show that the acceleration of the particle is proportional to the position coordinate. What is the constant of proportionality?

A particle moves along a coordinate line, its position at time t given by the function $x(t)=A e^{k \gamma}+B e^{-k t} . \quad(A>0 . B>0, k>0)$
(a) Find the times $t$ at which the particle is closest to the origin.
(b) Show that the acceleration of the particle is proportional to the position coordinate. What is the constant of proportionality?

Key Concepts

Proportionality in Differential Equations

In the context of differential equations, showing that the acceleration (the second derivative of position) is proportional to the position itself relates to identifying linear relationships. This indicates that the motion of the particle can be described by a second-order linear differential equation with constant coefficients, where the constant of proportionality reveals key characteristics of the system's dynamics.

Exponential Functions

Exponential functions are functions that involve constants raised to a power proportional to the variable. They frequently model growth and decay processes. In this problem, the position function is composed of two exponential terms whose behavior and rates dominate at different times.

Optimization (Finding Critical Points)

Optimization involves identifying points at which a function achieves its minimum or maximum values. This typically requires finding points where the first derivative equals zero and then using the second derivative test to confirm whether these points are minima or maxima.

Differentiation

Differentiation is a fundamental concept in calculus used to determine the rate at which a function changes. In this problem, it is used to compute the first derivative to find critical points where the function might attain a minimum or maximum, and the second derivative to verify the nature of these points.

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